P. 363 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36201-36300   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-mdl 36201*	Define the set of models of a formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mMdl = {𝑡 ∈ mFS ∣ [(mUV‘𝑡) / 𝑢][(mEx‘𝑡) / 𝑥][(mVL‘𝑡) / 𝑣][(mEval‘𝑡) / 𝑛][(mFresh‘𝑡) / 𝑓]((𝑢 ⊆ ((mTC‘𝑡) × V) ∧ 𝑓 ∈ (mFRel‘𝑡) ∧ 𝑛 ∈ (𝑢 ↑pm (𝑣 × (mEx‘𝑡)))) ∧ ∀𝑚 ∈ 𝑣 ((∀𝑒 ∈ 𝑥 (𝑛 “ {⟨𝑚, 𝑒⟩}) ⊆ (𝑢 “ {(1st ‘𝑒)}) ∧ ∀𝑦 ∈ (mVR‘𝑡)⟨𝑚, ((mVH‘𝑡)‘𝑦)⟩𝑛(𝑚‘𝑦) ∧ ∀𝑑∀ℎ∀𝑎(⟨𝑑, ℎ, 𝑎⟩ ∈ (mAx‘𝑡) → ((∀𝑦∀𝑧(𝑦𝑑𝑧 → (𝑚‘𝑦)𝑓(𝑚‘𝑧)) ∧ ℎ ⊆ (dom 𝑛 “ {𝑚})) → 𝑚dom 𝑛 𝑎))) ∧ (∀𝑠 ∈ ran (mSubst‘𝑡)∀𝑒 ∈ (mEx‘𝑡)∀𝑦(⟨𝑠, 𝑚⟩(mVSubst‘𝑡)𝑦 → (𝑛 “ {⟨𝑚, (𝑠‘𝑒)⟩}) = (𝑛 “ {⟨𝑦, 𝑒⟩})) ∧ ∀𝑝 ∈ 𝑣 ∀𝑒 ∈ 𝑥 ((𝑚 ↾ ((mVars‘𝑡)‘𝑒)) = (𝑝 ↾ ((mVars‘𝑡)‘𝑒)) → (𝑛 “ {⟨𝑚, 𝑒⟩}) = (𝑛 “ {⟨𝑝, 𝑒⟩})) ∧ ∀𝑦 ∈ 𝑢 ∀𝑒 ∈ 𝑥 ((𝑚 “ ((mVars‘𝑡)‘𝑒)) ⊆ (𝑓 “ {𝑦}) → (𝑛 “ {⟨𝑚, 𝑒⟩}) ⊆ (𝑓 “ {𝑦})))))}
Definition	df-musyn 36202*	Define the syntax typecode function for the model universe. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ ⟨((mSyn‘𝑡)‘(1st ‘𝑣)), (2nd ‘𝑣)⟩))
Definition	df-gmdl 36203*	Define the set of models of a grammatical formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mGMdl = {𝑡 ∈ (mGFS ∩ mMdl) ∣ (∀𝑐 ∈ (mTC‘𝑡)((mUV‘𝑡) “ {𝑐}) ⊆ ((mUV‘𝑡) “ {((mSyn‘𝑡)‘𝑐)}) ∧ ∀𝑣 ∈ (mUV‘𝑐)∀𝑤 ∈ (mUV‘𝑐)(𝑣(mFresh‘𝑡)𝑤 ↔ 𝑣(mFresh‘𝑡)((mUSyn‘𝑡)‘𝑤)) ∧ ∀𝑚 ∈ (mVL‘𝑡)∀𝑒 ∈ (mEx‘𝑡)((mEval‘𝑡) “ {⟨𝑚, 𝑒⟩}) = (((mEval‘𝑡) “ {⟨𝑚, ((mESyn‘𝑡)‘𝑒)⟩}) ∩ ((mUV‘𝑡) “ {(1st ‘𝑒)})))}
Definition	df-mitp 36204*	Define the interpretation function for a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mItp = (𝑡 ∈ V ↦ (𝑎 ∈ (mSA‘𝑡) ↦ (𝑔 ∈ X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)}) ↦ (℩𝑥∃𝑚 ∈ (mVL‘𝑡)(𝑔 = (𝑚 ↾ ((mVars‘𝑡)‘𝑎)) ∧ 𝑥 = (𝑚(mEval‘𝑡)𝑎))))))
Definition	df-mfitp 36205*	Define a function that produces the evaluation function, given the interpretation function for a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mFromItp = (𝑡 ∈ V ↦ (𝑓 ∈ X𝑎 ∈ (mSA‘𝑡)(((mUV‘𝑡) “ {((1st ‘𝑡)‘𝑎)}) ↑m X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)})) ↦ (℩𝑛 ∈ ((mUV‘𝑡) ↑pm ((mVL‘𝑡) × (mEx‘𝑡)))∀𝑚 ∈ (mVL‘𝑡)(∀𝑣 ∈ (mVR‘𝑡)⟨𝑚, ((mVH‘𝑡)‘𝑣)⟩𝑛(𝑚‘𝑣) ∧ ∀𝑒∀𝑎∀𝑔(𝑒(mST‘𝑡)⟨𝑎, 𝑔⟩ → ⟨𝑚, 𝑒⟩𝑛(𝑓‘(𝑖 ∈ ((mVars‘𝑡)‘𝑎) ↦ (𝑚𝑛(𝑔‘((mVH‘𝑡)‘𝑖)))))) ∧ ∀𝑒 ∈ (mEx‘𝑡)(𝑛 “ {⟨𝑚, 𝑒⟩}) = ((𝑛 “ {⟨𝑚, ((mESyn‘𝑡)‘𝑒)⟩}) ∩ ((mUV‘𝑡) “ {(1st ‘𝑒)}))))))
21.6.16  Splitting fields
Syntax	ccpms 36206	Completion of a metric space.
class cplMetSp
Syntax	chlb 36207	Embeddings for a direct limit.
class HomLimB
Syntax	chlim 36208	Direct limit structure.
class HomLim
Syntax	cpfl 36209	Polynomial extension field.
class polyFld
Syntax	csf1 36210	Splitting field for a single polynomial (auxiliary).
class splitFld1
Syntax	csf 36211	Splitting field for a finite set of polynomials.
class splitFld
Syntax	cpsl 36212	Splitting field for a sequence of polynomials.
class polySplitLim
Definition	df-cplmet 36213*	A function which completes the given metric space. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ cplMetSp = (𝑤 ∈ V ↦ ⦋((𝑤 ↑s ℕ) ↾s (Cau‘(dist‘𝑤))) / 𝑟⦌⦋(Base‘𝑟) / 𝑣⦌⦋{⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶((𝑔‘𝑗)(ball‘(dist‘𝑤))𝑥))} / 𝑒⦌((𝑟 /s 𝑒) sSet {⟨(dist‘ndx), {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝 ∈ 𝑣 ∃𝑞 ∈ 𝑣 ((𝑥 = [𝑝]𝑒 ∧ 𝑦 = [𝑞]𝑒) ∧ (𝑝 ∘f (dist‘𝑟)𝑞) ⇝ 𝑧)}⟩}))
Definition	df-homlimb 36214*	The input to this function is a sequence (on ℕ) of homomorphisms 𝐹(𝑛):𝑅(𝑛)⟶𝑅(𝑛 + 1). The resulting structure is the direct limit of the direct system so defined. This function returns the pair ⟨𝑆, 𝐺⟩ where 𝑆 is the terminal object and 𝐺 is a sequence of functions such that 𝐺(𝑛):𝑅(𝑛)⟶𝑆 and 𝐺(𝑛) = 𝐹(𝑛) ∘ 𝐺(𝑛 + 1). (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ HomLimB = (𝑓 ∈ V ↦ ⦋∪ 𝑛 ∈ ℕ ({𝑛} × dom (𝑓‘𝑛)) / 𝑣⦌⦋∩ {𝑠 ∣ (𝑠 Er 𝑣 ∧ (𝑥 ∈ 𝑣 ↦ ⟨((1st ‘𝑥) + 1), ((𝑓‘(1st ‘𝑥))‘(2nd ‘𝑥))⟩) ⊆ 𝑠)} / 𝑒⦌⟨(𝑣 / 𝑒), (𝑛 ∈ ℕ ↦ (𝑥 ∈ dom (𝑓‘𝑛) ↦ [⟨𝑛, 𝑥⟩]𝑒))⟩)
Definition	df-homlim 36215*	The input to this function is a sequence (on ℕ) of structures 𝑅(𝑛) and homomorphisms 𝐹(𝑛):𝑅(𝑛)⟶𝑅(𝑛 + 1). The resulting structure is the direct limit of the direct system so defined, and maintains any structures that were present in the original objects. TODO: generalize to directed sets? (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ HomLim = (𝑟 ∈ V, 𝑓 ∈ V ↦ ⦋( HomLimB ‘𝑓) / 𝑒⦌⦋(1st ‘𝑒) / 𝑣⦌⦋(2nd ‘𝑒) / 𝑔⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔‘𝑛), 𝑦 ∈ dom (𝑔‘𝑛) ↦ ⟨⟨((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)⟩, ((𝑔‘𝑛)‘(𝑥(+g‘(𝑟‘𝑛))𝑦))⟩)⟩, ⟨(.r‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔‘𝑛), 𝑦 ∈ dom (𝑔‘𝑛) ↦ ⟨⟨((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)⟩, ((𝑔‘𝑛)‘(𝑥(.r‘(𝑟‘𝑛))𝑦))⟩)⟩} ∪ {⟨(TopOpen‘ndx), {𝑠 ∈ 𝒫 𝑣 ∣ ∀𝑛 ∈ ℕ (◡(𝑔‘𝑛) “ 𝑠) ∈ (TopOpen‘(𝑟‘𝑛))}⟩, ⟨(dist‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom ((𝑔‘𝑛)‘𝑛), 𝑦 ∈ dom ((𝑔‘𝑛)‘𝑛) ↦ ⟨⟨((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)⟩, (𝑥(dist‘(𝑟‘𝑛))𝑦)⟩)⟩, ⟨(le‘ndx), ∪ 𝑛 ∈ ℕ (◡(𝑔‘𝑛) ∘ ((le‘(𝑟‘𝑛)) ∘ (𝑔‘𝑛)))⟩}))
Definition	df-plfl 36216*	Define the field extension that augments a field with the root of the given irreducible polynomial, and extends the norm if one exists and the extension is unique. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Thierry Arnoux and Steven Nguyen, 21-Jun-2025.)
⊢ polyFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ ⦋(Poly1‘𝑟) / 𝑠⦌⦋((RSpan‘𝑠)‘{𝑝}) / 𝑖⦌⦋(𝑐 ∈ (Base‘𝑟) ↦ [(𝑐( ·𝑠 ‘𝑠)(1r‘𝑠))](𝑠 ~QG 𝑖)) / 𝑓⦌⟨⦋(𝑠 /s (𝑠 ~QG 𝑖)) / 𝑡⦌((𝑡 toNrmGrp (℩𝑛 ∈ (AbsVal‘𝑡)(𝑛 ∘ 𝑓) = (norm‘𝑟))) sSet ⟨(le‘ndx), ⦋(𝑧 ∈ (Base‘𝑡) ↦ (℩𝑞 ∈ 𝑧 (𝑞(rem1p‘𝑟)𝑝) = 𝑞)) / 𝑔⦌(◡𝑔 ∘ ((le‘𝑠) ∘ 𝑔))⟩), 𝑓⟩)
Theorem	rexxfr3d 36217*	Transfer existential quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by SN, 20-Jun-2025.)
⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
Theorem	rexxfr3dALT 36218*	Longer proof of rexxfr3d 36217 using ax-11 2194 instead of ax-12 2213, without the disjoint variable condition 𝐴𝑥𝑦. (Contributed by SN, 19-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
Theorem	rspssbasd 36219	The span of a set of ring elements is a set of ring elements. (Contributed by SN, 19-Jun-2025.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐾‘𝐺) ⊆ 𝐵)
Theorem	ellcsrspsn 36220*	Membership in a left coset in a quotient of a ring by the span of a singleton (that is, by the ideal generated by an element). This characterization comes from eqglact 19304 and elrspsn 21434. (Contributed by SN, 19-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑈 = (𝑅 /s ∼ )    &   ⊢ 𝐼 = ((RSpan‘𝑅)‘{𝑀})    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (𝑋 = [𝑥] ∼ ∧ 𝑋 = {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝑧 = (𝑥 + (𝑦 · 𝑀))}))
Theorem	ply1divalg3 36221*	Uniqueness of polynomial remainder: convert the subtraction in ply1divalg2 26364 to addition. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ + = (+g‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 + (𝑞 ∙ 𝐺))) < (𝐷‘𝐺))
Theorem	r1peuqusdeg1 36222*	Uniqueness of polynomial remainder in terms of a quotient structure in the sense of the right hand side of r1pid2 26387. (Contributed by SN, 21-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐼 = ((RSpan‘𝑃)‘{𝐹})    &   ⊢ 𝑇 = (𝑃 /s (𝑃 ~QG 𝐼))    &   ⊢ 𝑄 = (Base‘𝑇)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑁)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑄)    ⇒   ⊢ (𝜑 → ∃!𝑞 ∈ 𝑍 (𝐷‘𝑞) < (𝐷‘𝐹))
Definition	df-sfl1 36223*	Temporary construction for the splitting field of a polynomial. The inputs are a field 𝑟 and a polynomial 𝑝 that we want to split, along with a tuple 𝑗 in the same format as the output. The output is a tuple ⟨𝑆, 𝐹⟩ where 𝑆 is the splitting field and 𝐹 is an injective homomorphism from the original field 𝑟. The function works by repeatedly finding the smallest monic irreducible factor, and extending the field by that factor using the polyFld construction. We keep track of a total order in each of the splitting fields so that we can pick an element definably without needing global choice. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ splitFld1 = (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))))
Definition	df-sfl 36224*	Define the splitting field of a finite collection of polynomials, given a total ordered base field. The output is a tuple ⟨𝑆, 𝐹⟩ where 𝑆 is the totally ordered splitting field and 𝐹 is an injective homomorphism from the original field 𝑟. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ splitFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ (℩𝑥∃𝑓(𝑓 Isom < , (lt‘𝑟)((1...(♯‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(♯‘𝑝)))))
Definition	df-psl 36225*	Define the direct limit of an increasing sequence of fields produced by pasting together the splitting fields for each sequence of polynomials. That is, given a ring 𝑟, a strict order on 𝑟, and a sequence 𝑝:ℕ⟶(𝒫 𝑟 ∩ Fin) of finite sets of polynomials to split, we construct the direct limit system of field extensions by splitting one set at a time and passing the resulting construction to HomLim. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ polySplitLim = (𝑟 ∈ V, 𝑝 ∈ ((𝒫 (Base‘𝑟) ∩ Fin) ↑m ℕ) ↦ ⦋(1st ∘ seq0((𝑔 ∈ V, 𝑞 ∈ V ↦ ⦋(1st ‘𝑔) / 𝑒⦌⦋(1st ‘𝑒) / 𝑠⦌⦋(𝑠 splitFld ran (𝑥 ∈ 𝑞 ↦ (𝑥 ∘ (2nd ‘𝑔)))) / 𝑓⦌⟨𝑓, ((2nd ‘𝑔) ∘ (2nd ‘𝑓))⟩), (𝑝 ∪ {⟨0, ⟨⟨𝑟, ∅⟩, ( I ↾ (Base‘𝑟))⟩⟩}))) / 𝑓⦌((1st ∘ (𝑓 shift 1)) HomLim (2nd ∘ 𝑓)))
21.6.17  p-adic number fields
Syntax	czr 36226	Integral elements of a ring.
class ZRing
Syntax	cgf 36227	Galois finite field.
class GF
Syntax	cgfo 36228	Galois limit field.
class GF∞
Syntax	ceqp 36229	Equivalence relation for df-qp 36240.
class ~Qp
Syntax	crqp 36230	Equivalence relation representatives for df-qp 36240.
class /Qp
Syntax	cqp 36231	The set of 𝑝-adic rational numbers.
class Qp
Syntax	czp 36232	The set of 𝑝-adic integers. (Not to be confused with czn 21715.)
class Zp
Syntax	cqpa 36233	Algebraic completion of the 𝑝-adic rational numbers.
class _Qp
Syntax	ccp 36234	Metric completion of _Qp.
class Cp
Definition	df-zrng 36235	Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟)))
Definition	df-gf 36236*	Define the Galois finite field of order 𝑝↑𝑛. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ GF = (𝑝 ∈ ℙ, 𝑛 ∈ ℕ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(1st ‘(𝑟 splitFld {⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)})))
Definition	df-gfoo 36237*	Define the Galois field of order 𝑝↑+∞, as a direct limit of the Galois finite fields. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ GF∞ = (𝑝 ∈ ℙ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)})))
Definition	df-eqp 36238*	Define an equivalence relation on ℤ-indexed sequences of integers such that two sequences are equivalent iff the difference is equivalent to zero, and a sequence is equivalent to zero iff the sum Σ𝑘 ≤ 𝑛𝑓(𝑘)(𝑝↑𝑘) is a multiple of 𝑝↑(𝑛 + 1) for every 𝑛. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ ~Qp = (𝑝 ∈ ℙ ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (ℤ ↑m ℤ) ∧ ∀𝑛 ∈ ℤ Σ𝑘 ∈ (ℤ≥‘-𝑛)(((𝑓‘-𝑘) − (𝑔‘-𝑘)) / (𝑝↑(𝑘 + (𝑛 + 1)))) ∈ ℤ)})
Definition	df-rqp 36239*	There is a unique element of (ℤ ↑m (0...(𝑝 − 1))) ~Qp -equivalent to any element of (ℤ ↑m ℤ), if the sequences are zero for sufficiently large negative values; this function selects that element. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ /Qp = (𝑝 ∈ ℙ ↦ (~Qp ∩ ⦋{𝑓 ∈ (ℤ ↑m ℤ) ∣ ∃𝑥 ∈ ran ℤ≥(◡𝑓 “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑦⦌(𝑦 × (𝑦 ∩ (ℤ ↑m (0...(𝑝 − 1)))))))
Definition	df-qp 36240*	Define the 𝑝-adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 10-Oct-2021.)
⊢ Qp = (𝑝 ∈ ℙ ↦ ⦋{ℎ ∈ (ℤ ↑m (0...(𝑝 − 1))) ∣ ∃𝑥 ∈ ran ℤ≥(◡ℎ “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑏⦌(({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑓 ∘f + 𝑔)))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑛 ∈ ℤ ↦ Σ𝑘 ∈ ℤ ((𝑓‘𝑘) · (𝑔‘(𝑛 − 𝑘))))))⟩} ∪ {⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑏 ∧ Σ𝑘 ∈ ℤ ((𝑓‘-𝑘) · ((𝑝 + 1)↑-𝑘)) < Σ𝑘 ∈ ℤ ((𝑔‘-𝑘) · ((𝑝 + 1)↑-𝑘)))}⟩}) toNrmGrp (𝑓 ∈ 𝑏 ↦ if(𝑓 = (ℤ × {0}), 0, (𝑝↑-inf((◡𝑓 “ (ℤ ∖ {0})), ℝ, < ))))))
Definition	df-zp 36241	Define the 𝑝-adic integers, as a subset of the 𝑝-adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ Zp = (ZRing ∘ Qp)
Definition	df-qpa 36242*	Define the completion of the 𝑝-adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the 𝑛-th set the collection of polynomials with degree less than 𝑛 and with coefficients < (𝑝↑𝑛)). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial 𝑥↑(𝑝↑𝑛) − 𝑥, which is in the list. Thus, every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ _Qp = (𝑝 ∈ ℙ ↦ ⦋(Qp‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {𝑓 ∈ (Poly1‘𝑟) ∣ ((𝑟deg1𝑓) ≤ 𝑛 ∧ ∀𝑑 ∈ ran (coe1‘𝑓)(◡𝑑 “ (ℤ ∖ {0})) ⊆ (0...𝑛))})))
Definition	df-cp 36243	Define the metric completion of the algebraic completion of the 𝑝 -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ Cp = ( cplMetSp ∘ _Qp)
21.7  Mathbox for Filip Cernatescu I hope someone will enjoy solving (proving) the simple equations, inequalities, and calculations from this mathbox. I have proved these problems (theorems) using the Milpgame proof assistant. (It can be downloaded from https://us.metamath.org/other/milpgame/milpgame.html.)
Theorem	problem1 36244	Practice problem 1. Clues: 5p4e9 12422 3p2e5 12415 eqtri 2783 oveq1i 7423. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ ((3 + 2) + 4) = 9
Theorem	problem2 36245	Practice problem 2. Clues: oveq12i 7425 adddiri 11246 add4i 11459 mulcli 11240 recni 11247 2re 12339 3eqtri 2787 10re 12759 5re 12352 1re 11232 4re 12349 eqcomi 2769 5p4e9 12422 oveq1i 7423 df-3 12328. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Revised by AV, 9-Sep-2021.) (Proof modification is discouraged.)
⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9)
Theorem	problem3 36246	Practice problem 3. Clues: eqcomi 2769 eqtri 2783 subaddrii 11571 recni 11247 4re 12349 3re 12345 1re 11232 df-4 12329 addcomi 11425. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ (𝐴 + 3) = 4    ⇒   ⊢ 𝐴 = 1
Theorem	problem4 36247	Practice problem 4. Clues: pm3.2i 476 eqcomi 2769 eqtri 2783 subaddrii 11571 recni 11247 7re 12358 6re 12355 ax-1cn 11182 df-7 12332 ax-mp 5 oveq1i 7423 3cn 12346 2cn 12340 df-3 12328 mullidi 11238 subdiri 11688 mp3an 1490 mulcli 11240 subadd23 11493 oveq2i 7424 oveq12i 7425 3t2e6 12430 mulcomi 11241 subcli 11558 biimpri 231 subadd2i 11570. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 + 𝐵) = 3    &   ⊢ ((3 · 𝐴) + (2 · 𝐵)) = 7    ⇒   ⊢ (𝐴 = 1 ∧ 𝐵 = 2)
Theorem	problem5 36248	Practice problem 5. Clues: 3brtr3i 5134 mpbi 233 breqtri 5130 ltaddsubi 11799 remulcli 11249 2re 12339 3re 12345 9re 12364 eqcomi 2769 mvlladdi 11500 3cn 6cn 12356 eqtr3i 2785 6p3e9 12424 addcomi 11425 ltdiv1ii 12168 6re 12355 nngt0i 12299 2nn 12338 divcan3i 11985 recni 11247 2cn 12340 2ne0 12371 mpbir 234 eqtri 2783 mulcomi 11241 3t2e6 12430 divmuli 11993. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℝ    &   ⊢ ((2 · 𝐴) + 3) < 9    ⇒   ⊢ 𝐴 < 3
Theorem	quad3 36249	Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.)
⊢ 𝑋 ∈ ℂ    &   ⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0    ⇒   ⊢ (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)))
21.8  Mathbox for Paul Chapman
21.8.1  Real and complex numbers (cont.)
Theorem	climuzcnv 36250*	Utility lemma to convert between 𝑚 ≤ 𝑘 and 𝑘 ∈ (ℤ≥‘𝑚) in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.)
⊢ (𝑚 ∈ ℕ → ((𝑘 ∈ (ℤ≥‘𝑚) → 𝜑) ↔ (𝑘 ∈ ℕ → (𝑚 ≤ 𝑘 → 𝜑))))
Theorem	sinccvglem 36251*	((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.)
⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0}))    &   ⊢ (𝜑 → 𝐹 ⇝ 0)    &   ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))    &   ⊢ 𝐻 = (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3)))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1)
Theorem	sinccvg 36252*	((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝐹 ⇝ 0) → ((𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥)) ∘ 𝐹) ⇝ 1)
Theorem	circum 36253*	The circumference of a circle of radius 𝑅, defined as the limit as 𝑛 ⇝ +∞ of the perimeter of an inscribed n-sided isogons, is ((2 · π) · 𝑅). (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
⊢ 𝐴 = ((2 · π) / 𝑛)    &   ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ ((2 · 𝑛) · (𝑅 · (sin‘(𝐴 / 2)))))    &   ⊢ 𝑅 ∈ ℝ    ⇒   ⊢ 𝑃 ⇝ ((2 · π) · 𝑅)
21.8.2  Miscellaneous theorems
Theorem	elfzm12 36254	Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ (𝑁 ∈ ℕ → (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ (1...𝑁)))
Theorem	nn0seqcvg 36255*	A strictly-decreasing nonnegative integer sequence with initial term 𝑁 reaches zero by the 𝑁 th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011.)
⊢ 𝐹:ℕ0⟶ℕ0    &   ⊢ 𝑁 = (𝐹‘0)    &   ⊢ (𝑘 ∈ ℕ0 → ((𝐹‘(𝑘 + 1)) ≠ 0 → (𝐹‘(𝑘 + 1)) < (𝐹‘𝑘)))    ⇒   ⊢ (𝐹‘𝑁) = 0
Theorem	lediv2aALT 36256	Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))
Theorem	abs2sqlei 36257	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2))
Theorem	abs2sqlti 36258	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2))
Theorem	abs2sqle 36259	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2)))
Theorem	abs2sqlt 36260	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2)))
Theorem	abs2difi 36261	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))
Theorem	abs2difabsi 36262	Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵))
21.9  Mathbox for Hongxiu Chen
Theorem	2thALT 36263	Alternate proof of 2th 267. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	orbi2iALT 36264	Alternate proof of orbi2i 926. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))
Theorem	pm3.48ALT 36265	Alternate proof of pm3.48 978. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃)))
Theorem	3jcadALT 36266	Alternate proof of 3jcad 1147. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) Use 3jcad instead. (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))
21.10  Mathbox for Adrian Ducourtial
21.10.1  Propositional calculus
Theorem	currybi 36267	Biconditional version of Curry's paradox. If some proposition 𝜑 amounts to the self-referential statement "This very statement is equivalent to 𝜓", then 𝜓 is true. See bj-currypara 37260 in BJ's mathbox for the classical version. (Contributed by Adrian Ducourtial, 18-Mar-2025.)
⊢ ((𝜑 ↔ (𝜑 ↔ 𝜓)) → 𝜓)
Theorem	antnest 36268	Suppose 𝜑, 𝜓 are distinct atomic propositional formulas, and let Γ be the smallest class of formulas for which ⊤ ∈ Γ and (𝜒 → 𝜑), (𝜒 → 𝜓) ∈ Γ for 𝜒 ∈ Γ. The present theorem is then an element of Γ, and the implications occurring in the theorem are in one-to-one correspondence with the formulas in Γ up to logical equivalence. In particular, the theorem itself is equivalent to ⊤ ∈ Γ. (Contributed by Adrian Ducourtial, 2-Oct-2025.)
⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
Theorem	antnestlaw3lem 36269	Lemma for antnestlaw3 36272. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ (¬ (((𝜑 → 𝜓) → 𝜒) → 𝜒) → ¬ (((𝜑 → 𝜒) → 𝜓) → 𝜓))
Theorem	antnestlaw1 36270	A law of nested antecedents. The converse direction is a subschema of pm2.27 43. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜓) ↔ (𝜑 → 𝜓))
Theorem	antnestlaw2 36271	A law of nested antecedents. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜒))
Theorem	antnestlaw3 36272	A law of nested antecedents. Compare with looinv 206. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ ((((𝜑 → 𝜓) → 𝜒) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜓))
Theorem	antnestALT 36273	Alternative proof of antnest 36268 from the valid schema ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) using laws of nested antecedents. Our proof uses only the laws antnestlaw1 36270 and antnestlaw3 36272. (Contributed by Adrian Ducourtial, 5-Dec-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
21.10.2  Clone theory
Syntax	ccloneop 36274	Syntax for the function of the class of operations on a set.
class CloneOp
Definition	df-cloneop 36275*	Define the function that sends a set to the class of clone-theoretic operations on the set. For convenience, we take an operation on 𝑎 to be a function on finite sequences of elements of 𝑎 (rather than tuples) with values in 𝑎. Following line 6 of [Szendrei] p. 11, the arity 𝑛 of an operation (here, the length of the sequences at which the operation is defined) is always finite and nonzero, whence 𝑛 is taken to be a nonzero finite ordinal. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ CloneOp = (𝑎 ∈ V ↦ {𝑥 ∣ ∃𝑛 ∈ (ω ∖ 1o)𝑥 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛))})
Syntax	cprj 36276	Syntax for the function of projections on sets.
class prj
Definition	df-prj 36277*	Define the function that, for a set 𝑎, arity 𝑛, and index 𝑖, returns the 𝑖-th 𝑛-ary projection on 𝑎. This is the 𝑛-ary operation on 𝑎 that, for any sequence of 𝑛 elements of 𝑎, returns the element having index 𝑖. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ prj = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑖 ∈ 𝑛 ↦ (𝑥 ∈ (𝑎 ↑m 𝑛) ↦ (𝑥‘𝑖))))
Syntax	csuppos 36278	Syntax for the function of superpositions.
class suppos
Definition	df-suppos 36279*	Define the function that, when given an 𝑛-ary operation 𝑓 and 𝑛 many 𝑚-ary operations (𝑔‘∅), ..., (𝑔‘∪ 𝑛), returns the superposition of 𝑓 with the (𝑔‘𝑖), itself another 𝑚-ary operation on 𝑎. Given 𝑥 (a sequence of 𝑚 arguments in 𝑎), the superposition effectively applies each of the (𝑔‘𝑖) to 𝑥, then applies 𝑓 to the resulting sequence of 𝑛 function values. This can be seen as a generalized version of function composition; see paragraph 3 of [Szendrei] p. 11. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ suppos = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑚 ∈ (ω ∖ 1o) ↦ (𝑓 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛)), 𝑔 ∈ ((𝑎 ↑m (𝑎 ↑m 𝑚)) ↑m 𝑛) ↦ (𝑥 ∈ (𝑎 ↑m 𝑚) ↦ (𝑓‘(𝑖 ∈ 𝑛 ↦ ((𝑔‘𝑖)‘𝑥)))))))
21.11  Mathbox for Scott Fenton
21.11.1  ZFC Axioms in primitive form
Theorem	axextprim 36280	ax-ext 2732 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧))
Theorem	axrepprim 36281	ax-rep 5232 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧 ∈ 𝑥 → ¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧 ∈ 𝑥)))
Theorem	axunprim 36282	ax-un 7736 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ∀𝑦(¬ ∀𝑥(𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)
Theorem	axpowprim 36283	ax-pow 5330 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) → 𝑥 = 𝑦)
Theorem	axregprim 36284	ax-reg 9564 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
Theorem	axinfprim 36285	ax-inf 9617 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ (𝑦 ∈ 𝑥 → ¬ ∀𝑦(𝑦 ∈ 𝑥 → ¬ ∀𝑧(𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥))))
Theorem	axacprim 36286	ax-ac 10461 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))
21.11.2  Untangled classes
Theorem	untelirr 36287*	We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 36369). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)
Theorem	untuni 36288*	The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥)
Theorem	untsucf 36289*	If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦)
Theorem	unt0 36290	The empty set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ 𝑥
Theorem	untint 36291*	If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦)
Theorem	efrunt 36292*	If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥)
Theorem	untangtr 36293*	A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.)
⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))
21.11.3  Extra propositional calculus theorems
Theorem	3jaodd 36294	Double deduction form of 3jaoi 1454. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))    &   ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃 ∨ 𝜏) → 𝜂)))
Theorem	3orit 36295	Closed form of 3ori 1451. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒))
Theorem	biimpexp 36296	A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜒)))
21.11.4  Misc. Useful Theorems
Theorem	nepss 36297	Two classes are unequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵))
Theorem	3ccased 36298	Triple disjunction form of ccased 1054. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝜑 → ((𝜒 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜎) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜎) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜎) → 𝜓))    ⇒   ⊢ (𝜑 → (((𝜒 ∨ 𝜃 ∨ 𝜏) ∧ (𝜂 ∨ 𝜁 ∨ 𝜎)) → 𝜓))
Theorem	dfso3 36299*	Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
⊢ (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
Theorem	brtpid1 36300	A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵