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Theorem goelel3xp 34963
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 34962 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
2 peano1 7898 . . . 4 ∅ ∈ ω
32a1i 11 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ ω)
4 opelxpi 5717 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
53, 4opelxpd 5719 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ (ω × (ω × ω)))
61, 5eqeltrd 2828 1 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ (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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