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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 32594 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) | |
2 | goel 32594 | . . 3 ⊢ ((𝑀 ∈ ω ∧ 𝑁 ∈ ω) → (𝑀∈𝑔𝑁) = 〈∅, 〈𝑀, 𝑁〉〉) | |
3 | 1, 2 | eqeqan12rd 2840 | . 2 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ 〈∅, 〈𝐼, 𝐽〉〉 = 〈∅, 〈𝑀, 𝑁〉〉)) |
4 | 0ex 5211 | . . . 4 ⊢ ∅ ∈ V | |
5 | opex 5356 | . . . 4 ⊢ 〈𝐼, 𝐽〉 ∈ V | |
6 | 4, 5 | opth 5368 | . . 3 ⊢ (〈∅, 〈𝐼, 𝐽〉〉 = 〈∅, 〈𝑀, 𝑁〉〉 ↔ (∅ = ∅ ∧ 〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉)) |
7 | eqid 2821 | . . . . 5 ⊢ ∅ = ∅ | |
8 | 7 | biantrur 533 | . . . 4 ⊢ (〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉 ↔ (∅ = ∅ ∧ 〈𝐼, 𝐽〉 = 〈𝑀, 𝑁〉)) |
9 | opthg 5369 | . . . . 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 1537 ∈ wcel 2114 ∅c0 4291 〈cop 4573 (class class class)co 7156 ωcom 7580 ∈𝑔cgoe 32580 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-goel 32587 |
This theorem is referenced by: satfv0 32605 satfv0fun 32618 |
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