Theorem goeleq12bg 34639

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 34637	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
2		goel 34637	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝑁 ∈ ω) → (𝑀∈𝑔𝑁) = ⟨∅, ⟨𝑀, 𝑁⟩⟩)
3	1, 2	eqeqan12rd 2746	. 2 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ ⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩))
4		0ex 5307	. . . 4 ⊢ ∅ ∈ V
5		opex 5464	. . . 4 ⊢ ⟨𝐼, 𝐽⟩ ∈ V
6	4, 5	opth 5476	. . 3 ⊢ (⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩ ↔ (∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩))
7		eqid 2731	. . . . 5 ⊢ ∅ = ∅
8	7	biantrur 530	. . . 4 ⊢ (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩))
9		opthg 5477	. . . . 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 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∅c0 4322  ⟨cop 4634  (class class class)co 7412  ωcom 7859  ∈_𝑔cgoe 34623

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-goel 34630

This theorem is referenced by:  satfv0  34648  satfv0fun  34661