Theorem vvdifopab 34024

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		opabid 4982	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
2	1	notbii 310	. . . 4 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ¬ 𝜑)
3		opelvvdif 34023	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
4	3	el2v 33984	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5		opabid 4982	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ¬ 𝜑)
6	2, 4, 5	3bitr4i 292	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
7	6	gen2 1723	. 2 ⊢ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
8		relxp 5227	. . . 4 ⊢ Rel (V × V)
9		reldif 5238	. . . 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 5247	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
12		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑥(V × V)
13		nfopab1 4719	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
14	12, 13	nfdif 3731	. . . 4 ⊢ Ⅎ𝑥((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15		nfopab1 4719	. . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
16		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑦(V × V)
17		nfopab2 4720	. . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
18	16, 17	nfdif 3731	. . . 4 ⊢ Ⅎ𝑦((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19		nfopab2 4720	. . . 4 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
20	14, 15, 18, 19	eqrelf 34020	. . 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 708	. 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 221	1 ⊢ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571  ⟨cop 4183  {copab 4712   × cxp 5112  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)