Theorem vvdifopab 39337

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 5501	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
2	1	notbii 323	. . . 4 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ¬ 𝜑)
3		opelvvdif 39336	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
4	3	el2v 3460	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5		opabidw 5501	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ¬ 𝜑)
6	2, 4, 5	3bitr4i 306	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
7	6	gen2 1828	. 2 ⊢ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
8		relxp 5672	. . . 4 ⊢ Rel (V × V)
9		reldif 5797	. . . 4 ⊢ (Rel (V × V) → Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
10	8, 9	ax-mp 5	. . 3 ⊢ Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
11		relopabv 5803	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
12		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥(V × V)
13		nfopab1 5177	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
14	12, 13	nfdif 4068	. . . 4 ⊢ Ⅎ𝑥((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15		nfopab1 5177	. . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
16		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑦(V × V)
17		nfopab2 5178	. . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
18	16, 17	nfdif 4068	. . . 4 ⊢ Ⅎ𝑦((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19		nfopab2 5178	. . . 4 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
20	14, 15, 18, 19	eqrelf 39330	. . 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 705	. 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 234	1 ⊢ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1567   = wceq 1569   ∈ wcel 2146  Vcvv 3453   ∖ cdif 4047  ⟨cop 4837  {copab 5169   × cxp 5652  Rel wrel 5659

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-opab 5170  df-xp 5660  df-rel 5661

This theorem is used by:  dfssr2  39651