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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 32613 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) | |
2 | goel 32613 | . . 3 ⊢ ((𝑀 ∈ ω ∧ 𝑁 ∈ ω) → (𝑀∈𝑔𝑁) = 〈∅, 〈𝑀, 𝑁〉〉) | |
3 | 1, 2 | eqeqan12rd 2839 | . 2 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ 〈∅, 〈𝐼, 𝐽〉〉 = 〈∅, 〈𝑀, 𝑁〉〉)) |
4 | 0ex 5204 | . . . 4 ⊢ ∅ ∈ V | |
5 | opex 5349 | . . . 4 ⊢ 〈𝐼, 𝐽〉 ∈ V | |
6 | 4, 5 | opth 5361 | . . 3 ⊢ (〈∅, 〈𝐼, 𝐽〉〉 = 〈∅, 〈𝑀, 𝑁〉〉 ↔ (∅ = ∅ ∧ 〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉)) |
7 | eqid 2820 | . . . . 5 ⊢ ∅ = ∅ | |
8 | 7 | biantrur 533 | . . . 4 ⊢ (〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉 ↔ (∅ = ∅ ∧ 〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉)) |
9 | opthg 5362 | . . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉 ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁))) | |
10 | 9 | adantl 484 | . . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → (〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉 ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁))) |
11 | 8, 10 | syl5bbr 287 | . . 3 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((∅ = ∅ ∧ 〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁))) |
12 | 6, 11 | syl5bb 285 | . 2 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → (〈∅, 〈𝐼, 𝐽〉〉 = 〈∅, 〈𝑀, 𝑁〉〉 ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁))) |
13 | 3, 12 | bitrd 281 | 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∅c0 4284 〈cop 4566 (class class class)co 7149 ωcom 7573 ∈𝑔cgoe 32599 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-goel 32606 |
This theorem is referenced by: satfv0 32624 satfv0fun 32637 |
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