MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xp1en Structured version   Visualization version   GIF version

Theorem xp1en 7993
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 7520 . . 3 1𝑜 = {∅}
21xpeq2i 5098 . 2 (𝐴 × 1𝑜) = (𝐴 × {∅})
3 0ex 4752 . . 3 ∅ ∈ V
4 xpsneng 7992 . . 3 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
53, 4mpan2 706 . 2 (𝐴𝑉 → (𝐴 × {∅}) ≈ 𝐴)
62, 5syl5eqbr 4650 1 (𝐴𝑉 → (𝐴 × 1𝑜) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Vcvv 3186  c0 3893  {csn 4150   class class class wbr 4615   × cxp 5074  1𝑜c1o 7501  cen 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-suc 5690  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-1o 7508  df-en 7903
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator