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Theorem xpsnen 6517
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 4011 . . 3 {𝐵} ∈ V
41, 3xpex 4541 . 2 (𝐴 × {𝐵}) ∈ V
5 elxp 4445 . . 3 (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})))
6 inteq 3686 . . . . . . . 8 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥, 𝑧⟩)
76inteqd 3688 . . . . . . 7 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥, 𝑧⟩)
8 vex 2622 . . . . . . . 8 𝑥 ∈ V
9 vex 2622 . . . . . . . 8 𝑧 ∈ V
108, 9op1stb 4290 . . . . . . 7 𝑥, 𝑧⟩ = 𝑥
117, 10syl6eq 2136 . . . . . 6 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥)
1211, 8syl6eqel 2178 . . . . 5 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 ∈ V)
1312adantr 270 . . . 4 ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) → 𝑦 ∈ V)
1413exlimivv 1824 . . 3 (∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) → 𝑦 ∈ V)
155, 14sylbi 119 . 2 (𝑦 ∈ (𝐴 × {𝐵}) → 𝑦 ∈ V)
168, 2opex 4047 . . 3 𝑥, 𝐵⟩ ∈ V
1716a1i 9 . 2 (𝑥𝐴 → ⟨𝑥, 𝐵⟩ ∈ V)
18 eqvisset 2629 . . . . 5 (𝑥 = 𝑦 𝑦 ∈ V)
19 ancom 262 . . . . . . . . . . 11 (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
20 anass 393 . . . . . . . . . . 11 (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})))
21 velsn 3458 . . . . . . . . . . . 12 (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
2221anbi1i 446 . . . . . . . . . . 11 ((𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
2319, 20, 223bitr3i 208 . . . . . . . . . 10 ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
2423exbii 1541 . . . . . . . . 9 (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ ∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
25 opeq2 3618 . . . . . . . . . . . 12 (𝑧 = 𝐵 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝐵⟩)
2625eqeq2d 2099 . . . . . . . . . . 11 (𝑧 = 𝐵 → (𝑦 = ⟨𝑥, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝐵⟩))
2726anbi1d 453 . . . . . . . . . 10 (𝑧 = 𝐵 → ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
282, 27ceqsexv 2658 . . . . . . . . 9 (∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴))
29 inteq 3686 . . . . . . . . . . . . . 14 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑦 = 𝑥, 𝐵⟩)
3029inteqd 3688 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑦 = 𝑥, 𝐵⟩)
318, 2op1stb 4290 . . . . . . . . . . . . 13 𝑥, 𝐵⟩ = 𝑥
3230, 31syl6req 2137 . . . . . . . . . . . 12 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑥 = 𝑦)
3332pm4.71ri 384 . . . . . . . . . . 11 (𝑦 = ⟨𝑥, 𝐵⟩ ↔ (𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩))
3433anbi1i 446 . . . . . . . . . 10 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ ((𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥𝐴))
35 anass 393 . . . . . . . . . 10 (((𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3634, 35bitri 182 . . . . . . . . 9 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3724, 28, 363bitri 204 . . . . . . . 8 (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3837exbii 1541 . . . . . . 7 (∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ ∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
395, 38bitri 182 . . . . . 6 (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
40 opeq1 3617 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝐵⟩ = ⟨ 𝑦, 𝐵⟩)
4140eqeq2d 2099 . . . . . . . 8 (𝑥 = 𝑦 → (𝑦 = ⟨𝑥, 𝐵⟩ ↔ 𝑦 = ⟨ 𝑦, 𝐵⟩))
42 eleq1 2150 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴 𝑦𝐴))
4341, 42anbi12d 457 . . . . . . 7 (𝑥 = 𝑦 → ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4443ceqsexgv 2744 . . . . . 6 ( 𝑦 ∈ V → (∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4539, 44syl5bb 190 . . . . 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 443 . . 3 ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = 𝑦) ↔ ((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦))
4832adantr 270 . . . . 5 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) → 𝑥 = 𝑦)
4948pm4.71i 383 . . . 4 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ∧ 𝑥 = 𝑦))
5043pm5.32ri 443 . . . 4 (((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ∧ 𝑥 = 𝑦) ↔ ((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦))
5149, 50bitr2i 183 . . 3 (((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴))
52 ancom 262 . . 3 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑥𝐴𝑦 = ⟨𝑥, 𝐵⟩))
5347, 51, 523bitri 204 . 2 ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = 𝑦) ↔ (𝑥𝐴𝑦 = ⟨𝑥, 𝐵⟩))
544, 1, 15, 17, 53en2i 6467 1 (𝐴 × {𝐵}) ≈ 𝐴
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  {csn 3441  cop 3444   cint 3683   class class class wbr 3837   × cxp 4426  cen 6435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-en 6438
This theorem is referenced by:  xpsneng  6518  endisj  6520
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