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Theorem goelel3xp 35308
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 35307 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
2 peano1 7923 . . . 4 ∅ ∈ ω
32a1i 11 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∅ ∈ ω)
4 opelxpi 5736 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
53, 4opelxpd 5738 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ (ω × (ω × ω)))
61, 5eqeltrd 2838 1 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ (ω × (ω × ω)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2103  c0 4347  cop 4654   × cxp 5697  (class class class)co 7445  ωcom 7899  𝑔cgoe 35293
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-ord 6397  df-on 6398  df-lim 6399  df-iota 6524  df-fun 6574  df-fv 6580  df-ov 7448  df-om 7900  df-goel 35300
This theorem is referenced by:  satfv0  35318  satf00  35334
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