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Theorem xpsnen 8999
Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
xpsnen.1 𝐴 ∈ V
xpsnen.2 𝐵 ∈ V
Assertion
Ref Expression
xpsnen (𝐴 × {𝐵}) ≈ 𝐴

Proof of Theorem xpsnen
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsnen.1 . . 3 𝐴 ∈ V
2 snex 5388 . . 3 {𝐵} ∈ V
31, 2xpex 7687 . 2 (𝐴 × {𝐵}) ∈ V
4 elxp 5656 . . 3 (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})))
5 inteq 4910 . . . . . . . 8 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥, 𝑧⟩)
65inteqd 4912 . . . . . . 7 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥, 𝑧⟩)
7 vex 3449 . . . . . . . 8 𝑥 ∈ V
8 vex 3449 . . . . . . . 8 𝑧 ∈ V
97, 8op1stb 5428 . . . . . . 7 𝑥, 𝑧⟩ = 𝑥
106, 9eqtrdi 2792 . . . . . 6 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥)
1110, 7eqeltrdi 2846 . . . . 5 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 ∈ V)
1211adantr 481 . . . 4 ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) → 𝑦 ∈ V)
1312exlimivv 1935 . . 3 (∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) → 𝑦 ∈ V)
144, 13sylbi 216 . 2 (𝑦 ∈ (𝐴 × {𝐵}) → 𝑦 ∈ V)
15 opex 5421 . . 3 𝑥, 𝐵⟩ ∈ V
1615a1i 11 . 2 (𝑥𝐴 → ⟨𝑥, 𝐵⟩ ∈ V)
17 eqvisset 3462 . . . . 5 (𝑥 = 𝑦 𝑦 ∈ V)
18 ancom 461 . . . . . . . . . . 11 (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
19 anass 469 . . . . . . . . . . 11 (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})))
20 velsn 4602 . . . . . . . . . . . 12 (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
2120anbi1i 624 . . . . . . . . . . 11 ((𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
2218, 19, 213bitr3i 300 . . . . . . . . . 10 ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
2322exbii 1850 . . . . . . . . 9 (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ ∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
24 xpsnen.2 . . . . . . . . . 10 𝐵 ∈ V
25 opeq2 4831 . . . . . . . . . . . 12 (𝑧 = 𝐵 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝐵⟩)
2625eqeq2d 2747 . . . . . . . . . . 11 (𝑧 = 𝐵 → (𝑦 = ⟨𝑥, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝐵⟩))
2726anbi1d 630 . . . . . . . . . 10 (𝑧 = 𝐵 → ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
2824, 27ceqsexv 3494 . . . . . . . . 9 (∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴))
29 inteq 4910 . . . . . . . . . . . . . 14 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑦 = 𝑥, 𝐵⟩)
3029inteqd 4912 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑦 = 𝑥, 𝐵⟩)
317, 24op1stb 5428 . . . . . . . . . . . . 13 𝑥, 𝐵⟩ = 𝑥
3230, 31eqtr2di 2793 . . . . . . . . . . . 12 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑥 = 𝑦)
3332pm4.71ri 561 . . . . . . . . . . 11 (𝑦 = ⟨𝑥, 𝐵⟩ ↔ (𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩))
3433anbi1i 624 . . . . . . . . . 10 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ ((𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥𝐴))
35 anass 469 . . . . . . . . . 10 (((𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3634, 35bitri 274 . . . . . . . . 9 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3723, 28, 363bitri 296 . . . . . . . 8 (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3837exbii 1850 . . . . . . 7 (∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ ∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
394, 38bitri 274 . . . . . 6 (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
40 opeq1 4830 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝐵⟩ = ⟨ 𝑦, 𝐵⟩)
4140eqeq2d 2747 . . . . . . . 8 (𝑥 = 𝑦 → (𝑦 = ⟨𝑥, 𝐵⟩ ↔ 𝑦 = ⟨ 𝑦, 𝐵⟩))
42 eleq1 2825 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴 𝑦𝐴))
4341, 42anbi12d 631 . . . . . . 7 (𝑥 = 𝑦 → ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4443ceqsexgv 3604 . . . . . 6 ( 𝑦 ∈ V → (∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4539, 44bitrid 282 . . . . 5 ( 𝑦 ∈ V → (𝑦 ∈ (𝐴 × {𝐵}) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4617, 45syl 17 . . . 4 (𝑥 = 𝑦 → (𝑦 ∈ (𝐴 × {𝐵}) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4746pm5.32ri 576 . . 3 ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = 𝑦) ↔ ((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦))
4832adantr 481 . . . . 5 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) → 𝑥 = 𝑦)
4948pm4.71i 560 . . . 4 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ∧ 𝑥 = 𝑦))
5043pm5.32ri 576 . . . 4 (((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ∧ 𝑥 = 𝑦) ↔ ((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦))
5149, 50bitr2i 275 . . 3 (((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴))
52 ancom 461 . . 3 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑥𝐴𝑦 = ⟨𝑥, 𝐵⟩))
5347, 51, 523bitri 296 . 2 ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = 𝑦) ↔ (𝑥𝐴𝑦 = ⟨𝑥, 𝐵⟩))
543, 1, 14, 16, 53en2i 8930 1 (𝐴 × {𝐵}) ≈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  Vcvv 3445  {csn 4586  cop 4592   cint 4907   class class class wbr 5105   × cxp 5631  cen 8880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-en 8884
This theorem is referenced by:  xpsneng  9000  endisj  9002  infxpenlem  9949  hashxplem  14333  rexpen  16110  heiborlem3  36272
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