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Theorem xpcoidgend 14323
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 4175 . . 3 (𝐴𝐵) = (𝐵𝐴)
2 xpcoidgend.1 . . 3 (𝜑 → (𝐴𝐵) ≠ ∅)
31, 2eqnetrrid 3088 . 2 (𝜑 → (𝐵𝐴) ≠ ∅)
43xpcogend 14322 1 (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wne 3013  cin 3932  c0 4288   × cxp 5546  ccom 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-co 5557
This theorem is referenced by:  xptrrel  14328  relexpxpnnidm  39926
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