Theorem vvdifopab 35523

Description: Ordered-pair class abstraction defined by a negation. (Contributed by Peter Mazsa, 25-Jun-2019.)

Assertion

Ref	Expression
vvdifopab	⊢ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem vvdifopab

Step	Hyp	Ref	Expression
1		opabidw 5414	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
2	1	notbii 322	. . . 4 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ¬ 𝜑)
3		opelvvdif 35522	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
4	3	el2v 3503	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5		opabidw 5414	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ¬ 𝜑)
6	2, 4, 5	3bitr4i 305	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
7	6	gen2 1797	. 2 ⊢ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
8		relxp 5575	. . . 4 ⊢ Rel (V × V)
9		reldif 5690	. . . 4 ⊢ (Rel (V × V) → Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
10	8, 9	ax-mp 5	. . 3 ⊢ Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
11		relopab 5698	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
12		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑥(V × V)
13		nfopab1 5137	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
14	12, 13	nfdif 4104	. . . 4 ⊢ Ⅎ𝑥((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15		nfopab1 5137	. . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
16		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑦(V × V)
17		nfopab2 5138	. . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
18	16, 17	nfdif 4104	. . . 4 ⊢ Ⅎ𝑦((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19		nfopab2 5138	. . . 4 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
20	14, 15, 18, 19	eqrelf 35519	. . 3 ⊢ ((Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ∧ Rel {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}) → (((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})))
21	10, 11, 20	mp2an 690	. 2 ⊢ (((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}))
22	7, 21	mpbir 233	1 ⊢ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1535   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ∖ cdif 3935  ⟨cop 4575  {copab 5130   × cxp 5555  Rel wrel 5562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-opab 5131  df-xp 5563  df-rel 5564

This theorem is referenced by:  dfssr2  35741