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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 35330 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) | |
| 2 | goel 35330 | . . 3 ⊢ ((𝑀 ∈ ω ∧ 𝑁 ∈ ω) → (𝑀∈𝑔𝑁) = 〈∅, 〈𝑀, 𝑁〉〉) | |
| 3 | 1, 2 | eqeqan12rd 2751 | . 2 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ 〈∅, 〈𝐼, 𝐽〉〉 = 〈∅, 〈𝑀, 𝑁〉〉)) |
| 4 | 0ex 5305 | . . . 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 2736 | . . . . 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 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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