Theorem negprop 38644

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 38641	. . 3 ⊢ PROP = setrecs((𝑦 ∈ V ↦ {𝑢 ∣ (∃𝑧 ∈ 𝑦 (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))}))
2		snexg 5397	. . 3 ⊢ (𝑥 ∈ PROP → {𝑥} ∈ V)
3		snssi 4745	. . 3 ⊢ (𝑥 ∈ PROP → {𝑥} ⊆ PROP)
4	1, 2, 3	setrec1 9972	. 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 4599	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
6	5	anbi1i 636	. . . . . . . . . 10 ⊢ ((𝑧 ∈ {𝑥} ∧ 𝑢 = (prop¬‘𝑧)) ↔ (𝑧 = 𝑥 ∧ 𝑢 = (prop¬‘𝑧)))
7	6	exbii 1880	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 ∈ {𝑥} ∧ 𝑢 = (prop¬‘𝑧)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑢 = (prop¬‘𝑧)))
8		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (prop¬‘𝑧) = (prop¬‘𝑥))
9	8	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑢 = (prop¬‘𝑧) ↔ 𝑢 = (prop¬‘𝑥)))
10	9	equsexvw 2037	. . . . . . . . 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 1959	. . . 4 ⊢ (𝑥 ∈ PROP → ∀𝑢(𝑢 = (prop¬‘𝑥) → (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))))
21		fvex 6898	. . . . 5 ⊢ (prop¬‘𝑥) ∈ V
22		elab6g 3622	. . . . 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 5392	. . . 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 38642	. . . . 5 ⊢ {𝑢 ∣ (∃𝑧 ∈ {𝑥} (𝑢 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ {𝑥}𝑢 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑢 = (propvar ‘𝑛))} ∈ V
33	30, 31, 32	fvmpt 6993	. . . 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 3909	1 ⊢ (𝑥 ∈ PROP → (prop¬‘𝑥) ∈ PROP)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∃wrex 3086  Vcvv 3450  {csn 4583   ↦ cmpt 5185  ‘cfv 6538  (class class class)co 7420  ℕcn 12335  propvar cpropvar 38634  prop¬cpropneg 38635  prop→cpropimp 38636  PROPcprop 38640

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642  ax-cnex 11256  ax-1cn 11258  ax-addcl 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769  df-scott 9929  df-setrecs 9965  df-nn 12336  df-prop 38641

This theorem is used by:  dfprop1  38646