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Theorem xp1en 8591
Description: One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
xp1en (𝐴𝑉 → (𝐴 × 1o) ≈ 𝐴)

Proof of Theorem xp1en
StepHypRef Expression
1 df1o2 8105 . . 3 1o = {∅}
21xpeq2i 5575 . 2 (𝐴 × 1o) = (𝐴 × {∅})
3 0ex 5202 . . 3 ∅ ∈ V
4 xpsneng 8590 . . 3 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
53, 4mpan2 687 . 2 (𝐴𝑉 → (𝐴 × {∅}) ≈ 𝐴)
62, 5eqbrtrid 5092 1 (𝐴𝑉 → (𝐴 × 1o) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  Vcvv 3492  c0 4288  {csn 4557   class class class wbr 5057   × cxp 5546  1oc1o 8084  cen 8494
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-pow 5257  ax-pr 5320  ax-un 7450
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-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  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-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-suc 6190  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-1o 8091  df-en 8498
This theorem is referenced by: (None)
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