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| Mirrors > Home > MPE Home > Th. List > Mathboxes > goel | Structured version Visualization version GIF version | ||
| Description: A "Godel-set of membership". The variables are identified by their indices (which are natural numbers), and the membership vi ∈ vj is coded as 〈∅, 〈𝑖, 𝑗〉〉. (Contributed by AV, 15-Sep-2023.) | 
| Ref | Expression | 
|---|---|
| goel | ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ov 7434 | . 2 ⊢ (𝐼∈𝑔𝐽) = (∈𝑔‘〈𝐼, 𝐽〉) | |
| 2 | df-goel 35345 | . . . 4 ⊢ ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ 〈∅, 𝑥〉) | |
| 3 | 2 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ 〈∅, 𝑥〉)) | 
| 4 | opeq2 4874 | . . . 4 ⊢ (𝑥 = 〈𝐼, 𝐽〉 → 〈∅, 𝑥〉 = 〈∅, 〈𝐼, 𝐽〉〉) | |
| 5 | 4 | adantl 481 | . . 3 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑥 = 〈𝐼, 𝐽〉) → 〈∅, 𝑥〉 = 〈∅, 〈𝐼, 𝐽〉〉) | 
| 6 | opelxpi 5722 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 〈𝐼, 𝐽〉 ∈ (ω × ω)) | |
| 7 | opex 5469 | . . . 4 ⊢ 〈∅, 〈𝐼, 𝐽〉〉 ∈ V | |
| 8 | 7 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 〈∅, 〈𝐼, 𝐽〉〉 ∈ V) | 
| 9 | 3, 5, 6, 8 | fvmptd 7023 | . 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∈𝑔‘〈𝐼, 𝐽〉) = 〈∅, 〈𝐼, 𝐽〉〉) | 
| 10 | 1, 9 | eqtrid 2789 | 1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ωcom 7887 ∈𝑔cgoe 35338 | 
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-goel 35345 | 
| This theorem is referenced by: goelel3xp 35353 goeleq12bg 35354 sat1el2xp 35384 fmla0xp 35388 fmlaomn0 35395 gonan0 35397 goaln0 35398 gonar 35400 goalr 35402 fmla0disjsuc 35403 satfv0fvfmla0 35418 sategoelfvb 35424 prv1n 35436 | 
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