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Theorem xpcoidgend 14980
Description: If two classes are not disjoint, then the composition of their Cartesian product with itself is idempotent. (Contributed by RP, 24-Dec-2019.)
Hypothesis
Ref Expression
xpcoidgend.1 (𝜑 → (𝐴𝐵) ≠ ∅)
Assertion
Ref Expression
xpcoidgend (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))

Proof of Theorem xpcoidgend
StepHypRef Expression
1 incom 4202 . . 3 (𝐴𝐵) = (𝐵𝐴)
2 xpcoidgend.1 . . 3 (𝜑 → (𝐴𝐵) ≠ ∅)
31, 2eqnetrrid 3006 . 2 (𝜑 → (𝐵𝐴) ≠ ∅)
43xpcogend 14979 1 (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wne 2930  cin 3946  c0 4325   × cxp 5680  ccom 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-co 5691
This theorem is referenced by:  xptrrel  14985  relexpxpnnidm  43370
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