Theorem goeleq12bg 35329

Description: Two "Godel-set of membership" codes for two variables are equal iff the two corresponding variables are equal. (Contributed by AV, 8-Oct-2023.)

Assertion

Ref	Expression
goeleq12bg	⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))

Proof of Theorem goeleq12bg

Step	Hyp	Ref	Expression
1		goel 35327	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
2		goel 35327	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝑁 ∈ ω) → (𝑀∈𝑔𝑁) = ⟨∅, ⟨𝑀, 𝑁⟩⟩)
3	1, 2	eqeqan12rd 2749	. 2 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ ⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩))
4		0ex 5287	. . . 4 ⊢ ∅ ∈ V
5		opex 5449	. . . 4 ⊢ ⟨𝐼, 𝐽⟩ ∈ V
6	4, 5	opth 5461	. . 3 ⊢ (⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩ ↔ (∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩))
7		eqid 2734	. . . . 5 ⊢ ∅ = ∅
8	7	biantrur 530	. . . 4 ⊢ (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩))
9		opthg 5462	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
10	9	adantl 481	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
11	8, 10	bitr3id 285	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
12	6, 11	bitrid 283	. 2 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → (⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩ ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
13	3, 12	bitrd 279	1 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∅c0 4313  ⟨cop 4612  (class class class)co 7413  ωcom 7869  ∈_𝑔cgoe 35313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-goel 35320

This theorem is referenced by:  satfv0  35338  satfv0fun  35351