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Mirrors > Home > MPE Home > Th. List > Mathboxes > goeleq12bg | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
goeleq12bg | ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | goel 35293 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) | |
2 | goel 35293 | . . 3 ⊢ ((𝑀 ∈ ω ∧ 𝑁 ∈ ω) → (𝑀∈𝑔𝑁) = 〈∅, 〈𝑀, 𝑁〉〉) | |
3 | 1, 2 | eqeqan12rd 2748 | . 2 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ 〈∅, 〈𝐼, 𝐽〉〉 = 〈∅, 〈𝑀, 𝑁〉〉)) |
4 | 0ex 5308 | . . . 4 ⊢ ∅ ∈ V | |
5 | opex 5467 | . . . 4 ⊢ 〈𝐼, 𝐽〉 ∈ V | |
6 | 4, 5 | opth 5479 | . . 3 ⊢ (〈∅, 〈𝐼, 𝐽〉〉 = 〈∅, 〈𝑀, 𝑁〉〉 ↔ (∅ = ∅ ∧ 〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉)) |
7 | eqid 2733 | . . . . 5 ⊢ ∅ = ∅ | |
8 | 7 | biantrur 530 | . . . 4 ⊢ (〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉 ↔ (∅ = ∅ ∧ 〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉)) |
9 | opthg 5480 | . . . . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1535 ∈ wcel 2104 ∅c0 4339 〈cop 4636 (class class class)co 7425 ωcom 7880 ∈𝑔cgoe 35279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6510 df-fun 6560 df-fv 6566 df-ov 7428 df-goel 35286 |
This theorem is referenced by: satfv0 35304 satfv0fun 35317 |
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