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Theorem xp1en 8315
 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 7839 . . 3 1o = {∅}
21xpeq2i 5369 . 2 (𝐴 × 1o) = (𝐴 × {∅})
3 0ex 5014 . . 3 ∅ ∈ V
4 xpsneng 8314 . . 3 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
53, 4mpan2 684 . 2 (𝐴𝑉 → (𝐴 × {∅}) ≈ 𝐴)
62, 5syl5eqbr 4908 1 (𝐴𝑉 → (𝐴 × 1o) ≈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2166  Vcvv 3414  ∅c0 4144  {csn 4397   class class class wbr 4873   × cxp 5340  1oc1o 7819   ≈ cen 8219 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-suc 5969  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-1o 7826  df-en 8223 This theorem is referenced by: (None)
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