Theorem vvdifopab 38516

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 5480	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
2	1	notbii 320	. . . 4 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ¬ 𝜑)
3		opelvvdif 38515	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
4	3	el2v 3449	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5		opabidw 5480	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ¬ 𝜑)
6	2, 4, 5	3bitr4i 303	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
7	6	gen2 1798	. 2 ⊢ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
8		relxp 5650	. . . 4 ⊢ Rel (V × V)
9		reldif 5772	. . . 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 5778	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
12		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑥(V × V)
13		nfopab1 5170	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
14	12, 13	nfdif 4083	. . . 4 ⊢ Ⅎ𝑥((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15		nfopab1 5170	. . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
16		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑦(V × V)
17		nfopab2 5171	. . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
18	16, 17	nfdif 4083	. . . 4 ⊢ Ⅎ𝑦((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19		nfopab2 5171	. . . 4 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
20	14, 15, 18, 19	eqrelf 38509	. . 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 693	. 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 231	1 ⊢ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1540   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∖ cdif 3900  ⟨cop 4588  {copab 5162   × cxp 5630  Rel wrel 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5638  df-rel 5639

This theorem is referenced by:  dfssr2  38830