Theorem negprop 38834

Description: The negation of a sentence of propositional calculus is a sentence of propositional calculus. (Contributed by Thomas van Maaren, 21-Aug-2026.)

Assertion

Ref	Expression
negprop	⊢ (𝑥 ∈ PROP → (prop¬‘𝑥) ∈ PROP)

Proof of Theorem negprop

Dummy variables 𝑤 𝑦 𝑧 𝑢 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prop 38831	. . 3 ⊢ PROP = setrecs((𝑦 ∈ V ↦ {𝑢 ∣ (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))}))
2		snexg 5402	. . 3 ⊢ (𝑥 ∈ PROP → {𝑥} ∈ V)
3		snssi 4746	. . 3 ⊢ (𝑥 ∈ PROP → {𝑥} ⊆ PROP)
4	1, 2, 3	setrec1 9996	. 2 ⊢ (𝑥 ∈ PROP → ((𝑦 ∈ V ↦ {𝑢 ∣ (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))})‘{𝑥}) ⊆ PROP)
5		velsn 4600	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
6	5	anbi1i 636	. . . . . . . . . 10 ⊢ ((𝑧 ∈ {𝑥} ∧ 𝑢 = (prop¬‘𝑧)) ↔ (𝑧 = 𝑥 ∧ 𝑢 = (prop¬‘𝑧)))
7	6	exbii 1881	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 ∈ {𝑥} ∧ 𝑢 = (prop¬‘𝑧)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑢 = (prop¬‘𝑧)))
8		fveq2 6905	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (prop¬‘𝑧) = (prop¬‘𝑥))
9	8	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑢 = (prop¬‘𝑧) ↔ 𝑢 = (prop¬‘𝑥)))
10	9	equsexvw 2040	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑢 = (prop¬‘𝑧)) ↔ 𝑢 = (prop¬‘𝑥))
11	7, 10	bitri 278	. . . . . . . 8 ⊢ (∃𝑧(𝑧 ∈ {𝑥} ∧ 𝑢 = (prop¬‘𝑧)) ↔ 𝑢 = (prop¬‘𝑥))
12	11	bilanri 512	. . . . . . 7 ⊢ ((𝑥 ∈ PROP ∧ 𝑢 = (prop¬‘𝑥)) → ∃𝑧(𝑧 ∈ {𝑥} ∧ 𝑢 = (prop¬‘𝑧)))
13		df-rex 3091	. . . . . . . 8 ⊢ (∃𝑧 ∈ {𝑥}𝑢 = (prop¬‘𝑧) ↔ ∃𝑧(𝑧 ∈ {𝑥} ∧ 𝑢 = (prop¬‘𝑧)))
14	13	biimpri 231	. . . . . . 7 ⊢ (∃𝑧(𝑧 ∈ {𝑥} ∧ 𝑢 = (prop¬‘𝑧)) → ∃𝑧 ∈ {𝑥}𝑢 = (prop¬‘𝑧))
15		orc 881	. . . . . . . . 9 ⊢ (𝑢 = (prop¬‘𝑧) → (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)))
16	15	reximi 3104	. . . . . . . 8 ⊢ (∃𝑧 ∈ {𝑥}𝑢 = (prop¬‘𝑧) → ∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)))
17	16	orcd 887	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑥}𝑢 = (prop¬‘𝑧) → (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛)))
18	12, 14, 17	3syl 19	. . . . . 6 ⊢ ((𝑥 ∈ PROP ∧ 𝑢 = (prop¬‘𝑥)) → (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛)))
19	18	ex 418	. . . . 5 ⊢ (𝑥 ∈ PROP → (𝑢 = (prop¬‘𝑥) → (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))))
20	19	alrimiv 1962	. . . 4 ⊢ (𝑥 ∈ PROP → ∀𝑢(𝑢 = (prop¬‘𝑥) → (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))))
21		fvex 6918	. . . . 5 ⊢ (prop¬‘𝑥) ∈ V
22		elab6g 3626	. . . . 5 ⊢ ((prop¬‘𝑥) ∈ V → ((prop¬‘𝑥) ∈ {𝑢 ∣ (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))} ↔ ∀𝑢(𝑢 = (prop¬‘𝑥) → (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛)))))
23	21, 22	ax-mp 5	. . . 4 ⊢ ((prop¬‘𝑥) ∈ {𝑢 ∣ (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))} ↔ ∀𝑢(𝑢 = (prop¬‘𝑥) → (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))))
24	20, 23	sylibr 237	. . 3 ⊢ (𝑥 ∈ PROP → (prop¬‘𝑥) ∈ {𝑢 ∣ (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))})
25		vsnex 5397	. . . 4 ⊢ {𝑥} ∈ V
26		rexeq 3319	. . . . . . . . 9 ⊢ (𝑦 = {𝑥} → (∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧) ↔ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)))
27	26	orbi2d 929	. . . . . . . 8 ⊢ (𝑦 = {𝑥} → ((𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ↔ (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧))))
28	27	rexeqbi1dv 3334	. . . . . . 7 ⊢ (𝑦 = {𝑥} → (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ↔ ∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧))))
29	28	orbi1d 930	. . . . . 6 ⊢ (𝑦 = {𝑥} → ((∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛)) ↔ (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))))
30	29	abbidv 2829	. . . . 5 ⊢ (𝑦 = {𝑥} → {𝑢 ∣ (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))} = {𝑢 ∣ (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))})
31		eqid 2763	. . . . 5 ⊢ (𝑦 ∈ V ↦ {𝑢 ∣ (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))}) = (𝑦 ∈ V ↦ {𝑢 ∣ (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))})
32	25	dfproplem 38832	. . . . 5 ⊢ {𝑢 ∣ (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))} ∈ V
33	30, 31, 32	fvmpt 7013	. . . 4 ⊢ ({𝑥} ∈ V → ((𝑦 ∈ V ↦ {𝑢 ∣ (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))})‘{𝑥}) = {𝑢 ∣ (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))})
34	25, 33	ax-mp 5	. . 3 ⊢ ((𝑦 ∈ V ↦ {𝑢 ∣ (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))})‘{𝑥}) = {𝑢 ∣ (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))}
35	24, 34	eleqtrrdi 2875	. 2 ⊢ (𝑥 ∈ PROP → (prop¬‘𝑥) ∈ ((𝑦 ∈ V ↦ {𝑢 ∣ (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))})‘{𝑥}))
36	4, 35	sseldd 3914	1 ⊢ (𝑥 ∈ PROP → (prop¬‘𝑥) ∈ PROP)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454  {csn 4588   ↦ cmpt 5189  ‘cfv 6555  (class class class)co 7440  ℕcn 12359  propvar cpropvar 38824  prop¬cpropneg 38825  prop→cpropimp 38826  PROPcprop 38830

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-reg 9610  ax-inf2 9666  ax-cnex 11280  ax-1cn 11282  ax-addcl 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-r1 9792  df-rank 9793  df-scott 9953  df-setrecs 9989  df-nn 12360  df-prop 38831

This theorem is used by:  dfprop1  38836