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Mirrors > Home > MPE Home > Th. List > Mathboxes > goelel3xp | Structured version Visualization version GIF version |
Description: A "Godel-set of membership" is a member of a doubled Cartesian product. (Contributed by AV, 16-Sep-2023.) |
Ref | Expression |
---|---|
goelel3xp | ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ (ω × (ω × ω))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | goel 35014 | . 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩) | |
2 | peano1 7892 | . . . 4 ⊢ ∅ ∈ ω | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ ω) |
4 | opelxpi 5709 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω)) | |
5 | 3, 4 | opelxpd 5711 | . 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ (ω × (ω × ω))) |
6 | 1, 5 | eqeltrd 2825 | 1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ (ω × (ω × ω))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ∅c0 4318 ⟨cop 4630 × cxp 5670 (class class class)co 7416 ωcom 7868 ∈𝑔cgoe 35000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-ord 6367 df-on 6368 df-lim 6369 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-om 7869 df-goel 35007 |
This theorem is referenced by: satfv0 35025 satf00 35041 |
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