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Theorem xpsnen2g 7997
Description: A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
Assertion
Ref Expression
xpsnen2g ((𝐴𝑉𝐵𝑊) → ({𝐴} × 𝐵) ≈ 𝐵)

Proof of Theorem xpsnen2g
StepHypRef Expression
1 snex 4869 . . . 4 {𝐴} ∈ V
2 xpcomeng 7996 . . . 4 (({𝐴} ∈ V ∧ 𝐵𝑊) → ({𝐴} × 𝐵) ≈ (𝐵 × {𝐴}))
31, 2mpan 705 . . 3 (𝐵𝑊 → ({𝐴} × 𝐵) ≈ (𝐵 × {𝐴}))
43adantl 482 . 2 ((𝐴𝑉𝐵𝑊) → ({𝐴} × 𝐵) ≈ (𝐵 × {𝐴}))
5 xpsneng 7989 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵 × {𝐴}) ≈ 𝐵)
65ancoms 469 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝐴}) ≈ 𝐵)
7 entr 7952 . 2 ((({𝐴} × 𝐵) ≈ (𝐵 × {𝐴}) ∧ (𝐵 × {𝐴}) ≈ 𝐵) → ({𝐴} × 𝐵) ≈ 𝐵)
84, 6, 7syl2anc 692 1 ((𝐴𝑉𝐵𝑊) → ({𝐴} × 𝐵) ≈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Vcvv 3186  {csn 4148   class class class wbr 4613   × cxp 5072  cen 7896
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1st 7113  df-2nd 7114  df-er 7687  df-en 7900
This theorem is referenced by:  unxpwdom2  8437  ackbij1lem8  8993  lgsquadlem1  25005  lgsquadlem2  25006
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