Theorem xpsneng 6969

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 4730	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 × {𝑦}) = (𝐴 × {𝑦}))
2		id 19	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
3	1, 2	breq12d 4095	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 × {𝑦}) ≈ 𝑥 ↔ (𝐴 × {𝑦}) ≈ 𝐴))
4		sneq 3677	. . . 4 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵})
5	4	xpeq2d 4740	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 × {𝑦}) = (𝐴 × {𝐵}))
6	5	breq1d 4092	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 × {𝑦}) ≈ 𝐴 ↔ (𝐴 × {𝐵}) ≈ 𝐴))
7		vex 2802	. . 3 ⊢ 𝑥 ∈ V
8		vex 2802	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	xpsnen 6968	. 2 ⊢ (𝑥 × {𝑦}) ≈ 𝑥
10	3, 6, 9	vtocl2g 2865	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {csn 3666   class class class wbr 4082   × cxp 4714   ≈ cen 6875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-en 6878

This theorem is referenced by:  xp1en  6970  xpsnen2g  6976  xpdom3m  6981  hashxp  11035  pwf1oexmid  16296