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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 34962 | . 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) | |
2 | peano1 7898 | . . . 4 ⊢ ∅ ∈ ω | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ ω) |
4 | opelxpi 5717 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 〈𝐼, 𝐽〉 ∈ (ω × ω)) | |
5 | 3, 4 | opelxpd 5719 | . 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 〈∅, 〈𝐼, 𝐽〉〉 ∈ (ω × (ω × ω))) |
6 | 1, 5 | eqeltrd 2828 | 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 4324 〈cop 4636 × cxp 5678 (class class class)co 7424 ωcom 7874 ∈𝑔cgoe 34948 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-ord 6375 df-on 6376 df-lim 6377 df-iota 6503 df-fun 6553 df-fv 6559 df-ov 7427 df-om 7875 df-goel 34955 |
This theorem is referenced by: satfv0 34973 satf00 34989 |
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