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Theorem goelel3xp 35318
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 35317 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
2 peano1 7929 . . . 4 ∅ ∈ ω
32a1i 11 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ ω)
4 opelxpi 5737 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
53, 4opelxpd 5739 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ (ω × (ω × ω)))
61, 5eqeltrd 2844 1 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ (ω × (ω × ω)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  c0 4352  cop 4654   × cxp 5698  (class class class)co 7450  ωcom 7905  𝑔cgoe 35303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-ord 6400  df-on 6401  df-lim 6402  df-iota 6527  df-fun 6577  df-fv 6583  df-ov 7453  df-om 7906  df-goel 35310
This theorem is referenced by:  satfv0  35328  satf00  35344
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