Theorem xpsneng 9095

Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)

Assertion

Ref	Expression
xpsneng	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)

Proof of Theorem xpsneng

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 5703	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 × {𝑦}) = (𝐴 × {𝑦}))
2		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
3	1, 2	breq12d 5161	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 × {𝑦}) ≈ 𝑥 ↔ (𝐴 × {𝑦}) ≈ 𝐴))
4		sneq 4641	. . . 4 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵})
5	4	xpeq2d 5719	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 × {𝑦}) = (𝐴 × {𝐵}))
6	5	breq1d 5158	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 × {𝑦}) ≈ 𝐴 ↔ (𝐴 × {𝐵}) ≈ 𝐴))
7		vex 3482	. . 3 ⊢ 𝑥 ∈ V
8		vex 3482	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	xpsnen 9094	. 2 ⊢ (𝑥 × {𝑦}) ≈ 𝑥
10	3, 6, 9	vtocl2g 3574	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {csn 4631   class class class wbr 5148   × cxp 5687   ≈ cen 8981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-en 8985

This theorem is referenced by:  xp1en  9096  xpsnen2g  9104  xpdom3  9109  disjen  9173  unxpdom2  9288  sucxpdom  9289  gchxpidm  10707  frlmiscvec  21887