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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 7156 | . 2 ⊢ (𝐼∈𝑔𝐽) = (∈𝑔‘〈𝐼, 𝐽〉) | |
2 | df-goel 32611 | . . . 4 ⊢ ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ 〈∅, 𝑥〉) | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ 〈∅, 𝑥〉)) |
4 | opeq2 4801 | . . . 4 ⊢ (𝑥 = 〈𝐼, 𝐽〉 → 〈∅, 𝑥〉 = 〈∅, 〈𝐼, 𝐽〉〉) | |
5 | 4 | adantl 484 | . . 3 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑥 = 〈𝐼, 𝐽〉) → 〈∅, 𝑥〉 = 〈∅, 〈𝐼, 𝐽〉〉) |
6 | opelxpi 5589 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 〈𝐼, 𝐽〉 ∈ (ω × ω)) | |
7 | opex 5353 | . . . 4 ⊢ 〈∅, 〈𝐼, 𝐽〉〉 ∈ V | |
8 | 7 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 〈∅, 〈𝐼, 𝐽〉〉 ∈ V) |
9 | 3, 5, 6, 8 | fvmptd 6772 | . 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∈𝑔‘〈𝐼, 𝐽〉) = 〈∅, 〈𝐼, 𝐽〉〉) |
10 | 1, 9 | syl5eq 2867 | 1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3493 ∅c0 4288 〈cop 4570 ↦ cmpt 5143 × cxp 5550 ‘cfv 6352 (class class class)co 7153 ωcom 7577 ∈𝑔cgoe 32604 |
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 5200 ax-nul 5207 ax-pr 5327 |
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 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-mpt 5144 df-id 5457 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-iota 6311 df-fun 6354 df-fv 6360 df-ov 7156 df-goel 32611 |
This theorem is referenced by: goelel3xp 32619 goeleq12bg 32620 sat1el2xp 32650 fmla0xp 32654 fmlaomn0 32661 gonan0 32663 goaln0 32664 gonar 32666 goalr 32668 fmla0disjsuc 32669 satfv0fvfmla0 32684 sategoelfvb 32690 prv1n 32702 |
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