Theorem negprop 38557

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 38554	. . 3 ⊢ PROP = setrecs((𝑦 ∈ V ↦ {𝑢 ∣ (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))}))
2		snexg 5398	. . 3 ⊢ (𝑥 ∈ PROP → {𝑥} ∈ V)
3		snssi 4746	. . 3 ⊢ (𝑥 ∈ PROP → {𝑥} ⊆ PROP)
4	1, 2, 3	setrec1 9929	. 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 6874	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (prop¬‘𝑧) = (prop¬‘𝑥))
9	8	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑢 = (prop¬‘𝑧) ↔ 𝑢 = (prop¬‘𝑥)))
10	9	equsexvw 2038	. . . . . . . . 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 3087	. . . . . . . 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 3100	. . . . . . . 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 1960	. . . 4 ⊢ (𝑥 ∈ PROP → ∀𝑢(𝑢 = (prop¬‘𝑥) → (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))))
21		fvex 6887	. . . . 5 ⊢ (prop¬‘𝑥) ∈ V
22		elab6g 3623	. . . . 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 5393	. . . 4 ⊢ {𝑥} ∈ V
26		rexeq 3315	. . . . . . . . 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 3330	. . . . . . 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 2826	. . . . 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 2760	. . . . 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 38555	. . . . 5 ⊢ {𝑢 ∣ (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))} ∈ V
33	30, 31, 32	fvmpt 6982	. . . 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 2871	. 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 3932	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 2145  {cab 2738  ∃wrex 3086  Vcvv 3450  {csn 4584   ↦ cmpt 5186  ‘cfv 6528  (class class class)co 7409  ℕcn 12290  propvar cpropvar 38547  prop¬cpropneg 38548  prop→cpropimp 38549  PROPcprop 38553

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-reg 9564  ax-inf2 9620  ax-cnex 11213  ax-1cn 11215  ax-addcl 11217

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-om 7862  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-r1 9746  df-rank 9747  df-scott 9886  df-setrecs 9922  df-nn 12291  df-prop 38554

This theorem is used by:  dfprop1  38559