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Theorem xpsnen 6681
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 xpsnen.2 . . . 4 𝐵 ∈ V
32snex 4077 . . 3 {𝐵} ∈ V
41, 3xpex 4622 . 2 (𝐴 × {𝐵}) ∈ V
5 elxp 4524 . . 3 (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})))
6 inteq 3742 . . . . . . . 8 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥, 𝑧⟩)
76inteqd 3744 . . . . . . 7 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥, 𝑧⟩)
8 vex 2661 . . . . . . . 8 𝑥 ∈ V
9 vex 2661 . . . . . . . 8 𝑧 ∈ V
108, 9op1stb 4367 . . . . . . 7 𝑥, 𝑧⟩ = 𝑥
117, 10syl6eq 2164 . . . . . 6 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥)
1211, 8syl6eqel 2206 . . . . 5 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 ∈ V)
1312adantr 272 . . . 4 ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) → 𝑦 ∈ V)
1413exlimivv 1850 . . 3 (∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) → 𝑦 ∈ V)
155, 14sylbi 120 . 2 (𝑦 ∈ (𝐴 × {𝐵}) → 𝑦 ∈ V)
168, 2opex 4119 . . 3 𝑥, 𝐵⟩ ∈ V
1716a1i 9 . 2 (𝑥𝐴 → ⟨𝑥, 𝐵⟩ ∈ V)
18 eqvisset 2668 . . . . 5 (𝑥 = 𝑦 𝑦 ∈ V)
19 ancom 264 . . . . . . . . . . 11 (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
20 anass 396 . . . . . . . . . . 11 (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})))
21 velsn 3512 . . . . . . . . . . . 12 (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
2221anbi1i 451 . . . . . . . . . . 11 ((𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
2319, 20, 223bitr3i 209 . . . . . . . . . 10 ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
2423exbii 1567 . . . . . . . . 9 (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ ∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
25 opeq2 3674 . . . . . . . . . . . 12 (𝑧 = 𝐵 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝐵⟩)
2625eqeq2d 2127 . . . . . . . . . . 11 (𝑧 = 𝐵 → (𝑦 = ⟨𝑥, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝐵⟩))
2726anbi1d 458 . . . . . . . . . 10 (𝑧 = 𝐵 → ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
282, 27ceqsexv 2697 . . . . . . . . 9 (∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴))
29 inteq 3742 . . . . . . . . . . . . . 14 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑦 = 𝑥, 𝐵⟩)
3029inteqd 3744 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑦 = 𝑥, 𝐵⟩)
318, 2op1stb 4367 . . . . . . . . . . . . 13 𝑥, 𝐵⟩ = 𝑥
3230, 31syl6req 2165 . . . . . . . . . . . 12 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑥 = 𝑦)
3332pm4.71ri 387 . . . . . . . . . . 11 (𝑦 = ⟨𝑥, 𝐵⟩ ↔ (𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩))
3433anbi1i 451 . . . . . . . . . 10 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ ((𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥𝐴))
35 anass 396 . . . . . . . . . 10 (((𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3634, 35bitri 183 . . . . . . . . 9 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3724, 28, 363bitri 205 . . . . . . . 8 (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3837exbii 1567 . . . . . . 7 (∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ ∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
395, 38bitri 183 . . . . . 6 (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
40 opeq1 3673 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝐵⟩ = ⟨ 𝑦, 𝐵⟩)
4140eqeq2d 2127 . . . . . . . 8 (𝑥 = 𝑦 → (𝑦 = ⟨𝑥, 𝐵⟩ ↔ 𝑦 = ⟨ 𝑦, 𝐵⟩))
42 eleq1 2178 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴 𝑦𝐴))
4341, 42anbi12d 462 . . . . . . 7 (𝑥 = 𝑦 → ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4443ceqsexgv 2786 . . . . . 6 ( 𝑦 ∈ V → (∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4539, 44syl5bb 191 . . . . 5 ( 𝑦 ∈ V → (𝑦 ∈ (𝐴 × {𝐵}) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4618, 45syl 14 . . . 4 (𝑥 = 𝑦 → (𝑦 ∈ (𝐴 × {𝐵}) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4746pm5.32ri 448 . . 3 ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = 𝑦) ↔ ((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦))
4832adantr 272 . . . . 5 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) → 𝑥 = 𝑦)
4948pm4.71i 386 . . . 4 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ∧ 𝑥 = 𝑦))
5043pm5.32ri 448 . . . 4 (((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ∧ 𝑥 = 𝑦) ↔ ((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦))
5149, 50bitr2i 184 . . 3 (((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴))
52 ancom 264 . . 3 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑥𝐴𝑦 = ⟨𝑥, 𝐵⟩))
5347, 51, 523bitri 205 . 2 ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = 𝑦) ↔ (𝑥𝐴𝑦 = ⟨𝑥, 𝐵⟩))
544, 1, 15, 17, 53en2i 6630 1 (𝐴 × {𝐵}) ≈ 𝐴
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  {csn 3495  cop 3498   cint 3739   class class class wbr 3897   × cxp 4505  cen 6598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-en 6601
This theorem is referenced by:  xpsneng  6682  endisj  6684
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