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Theorem xpcoidgend 13562
 Description: If two classes are not disjoint, then the composition of their cross-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 3767 . . 3 (𝐴𝐵) = (𝐵𝐴)
2 xpcoidgend.1 . . 3 (𝜑 → (𝐴𝐵) ≠ ∅)
31, 2syl5eqner 2857 . 2 (𝜑 → (𝐵𝐴) ≠ ∅)
43xpcogend 13561 1 (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ≠ wne 2780   ∩ cin 3539  ∅c0 3874   × cxp 5036   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-co 5047 This theorem is referenced by:  xptrrel  13567  relexpxpnnidm  37014
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