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Theorem goeleq12bg 35332
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)) → ((𝐼𝑔𝐽) = (𝑀𝑔𝑁) ↔ (𝐼 = 𝑀𝐽 = 𝑁)))

Proof of Theorem goeleq12bg
StepHypRef Expression
1 goel 35330 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
2 goel 35330 . . 3 ((𝑀 ∈ ω ∧ 𝑁 ∈ ω) → (𝑀𝑔𝑁) = ⟨∅, ⟨𝑀, 𝑁⟩⟩)
31, 2eqeqan12rd 2751 . 2 (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼𝑔𝐽) = (𝑀𝑔𝑁) ↔ ⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩))
4 0ex 5305 . . . 4 ∅ ∈ V
5 opex 5467 . . . 4 𝐼, 𝐽⟩ ∈ V
64, 5opth 5479 . . 3 (⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩ ↔ (∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩))
7 eqid 2736 . . . . 5 ∅ = ∅
87biantrur 530 . . . 4 (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩))
9 opthg 5480 . . . . 5 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
109adantl 481 . . . 4 (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
118, 10bitr3id 285 . . 3 (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩) ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
126, 11bitrid 283 . 2 (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → (⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩ ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
133, 12bitrd 279 1 (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼𝑔𝐽) = (𝑀𝑔𝑁) ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  c0 4332  cop 4630  (class class class)co 7429  ωcom 7883  𝑔cgoe 35316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6512  df-fun 6561  df-fv 6567  df-ov 7432  df-goel 35323
This theorem is referenced by:  satfv0  35341  satfv0fun  35354
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