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Theorem goelel3xp 35015
Description: A "Godel-set of membership" is a member of a doubled Cartesian product. (Contributed by AV, 16-Sep-2023.)
Assertion
Ref Expression
goelel3xp ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ (ω × (ω × ω)))

Proof of Theorem goelel3xp
StepHypRef Expression
1 goel 35014 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
2 peano1 7892 . . . 4 ∅ ∈ ω
32a1i 11 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ ω)
4 opelxpi 5709 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
53, 4opelxpd 5711 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ (ω × (ω × ω)))
61, 5eqeltrd 2825 1 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ (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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