Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpsnen GIF version

Theorem xpsnen 6326
 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 3965 . . 3 {𝐵} ∈ V
41, 3xpex 4481 . 2 (𝐴 × {𝐵}) ∈ V
5 elxp 4390 . . 3 (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})))
6 inteq 3646 . . . . . . . 8 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥, 𝑧⟩)
76inteqd 3648 . . . . . . 7 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥, 𝑧⟩)
8 vex 2577 . . . . . . . 8 𝑥 ∈ V
9 vex 2577 . . . . . . . 8 𝑧 ∈ V
108, 9op1stb 4237 . . . . . . 7 𝑥, 𝑧⟩ = 𝑥
117, 10syl6eq 2104 . . . . . 6 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 = 𝑥)
1211, 8syl6eqel 2144 . . . . 5 (𝑦 = ⟨𝑥, 𝑧⟩ → 𝑦 ∈ V)
1312adantr 265 . . . 4 ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) → 𝑦 ∈ V)
1413exlimivv 1792 . . 3 (∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) → 𝑦 ∈ V)
155, 14sylbi 118 . 2 (𝑦 ∈ (𝐴 × {𝐵}) → 𝑦 ∈ V)
168, 2opex 3994 . . 3 𝑥, 𝐵⟩ ∈ V
1716a1i 9 . 2 (𝑥𝐴 → ⟨𝑥, 𝐵⟩ ∈ V)
18 eqvisset 2582 . . . . 5 (𝑥 = 𝑦 𝑦 ∈ V)
19 ancom 257 . . . . . . . . . . 11 (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
20 anass 387 . . . . . . . . . . 11 (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})))
21 velsn 3420 . . . . . . . . . . . 12 (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
2221anbi1i 439 . . . . . . . . . . 11 ((𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
2319, 20, 223bitr3i 203 . . . . . . . . . 10 ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
2423exbii 1512 . . . . . . . . 9 (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ ∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)))
25 opeq2 3578 . . . . . . . . . . . 12 (𝑧 = 𝐵 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝐵⟩)
2625eqeq2d 2067 . . . . . . . . . . 11 (𝑧 = 𝐵 → (𝑦 = ⟨𝑥, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝐵⟩))
2726anbi1d 446 . . . . . . . . . 10 (𝑧 = 𝐵 → ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
282, 27ceqsexv 2610 . . . . . . . . 9 (∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥𝐴)) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴))
29 inteq 3646 . . . . . . . . . . . . . 14 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑦 = 𝑥, 𝐵⟩)
3029inteqd 3648 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑦 = 𝑥, 𝐵⟩)
318, 2op1stb 4237 . . . . . . . . . . . . 13 𝑥, 𝐵⟩ = 𝑥
3230, 31syl6req 2105 . . . . . . . . . . . 12 (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑥 = 𝑦)
3332pm4.71ri 378 . . . . . . . . . . 11 (𝑦 = ⟨𝑥, 𝐵⟩ ↔ (𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩))
3433anbi1i 439 . . . . . . . . . 10 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ ((𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥𝐴))
35 anass 387 . . . . . . . . . 10 (((𝑥 = 𝑦𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3634, 35bitri 177 . . . . . . . . 9 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3724, 28, 363bitri 199 . . . . . . . 8 (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ (𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
3837exbii 1512 . . . . . . 7 (∃𝑥𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝐴𝑧 ∈ {𝐵})) ↔ ∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
395, 38bitri 177 . . . . . 6 (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)))
40 opeq1 3577 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝐵⟩ = ⟨ 𝑦, 𝐵⟩)
4140eqeq2d 2067 . . . . . . . 8 (𝑥 = 𝑦 → (𝑦 = ⟨𝑥, 𝐵⟩ ↔ 𝑦 = ⟨ 𝑦, 𝐵⟩))
42 eleq1 2116 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴 𝑦𝐴))
4341, 42anbi12d 450 . . . . . . 7 (𝑥 = 𝑦 → ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4443ceqsexgv 2696 . . . . . 6 ( 𝑦 ∈ V → (∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴)) ↔ (𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴)))
4539, 44syl5bb 185 . . . . 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 436 . . 3 ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = 𝑦) ↔ ((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦))
4832adantr 265 . . . . 5 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) → 𝑥 = 𝑦)
4948pm4.71i 377 . . . 4 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ∧ 𝑥 = 𝑦))
5043pm5.32ri 436 . . . 4 (((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ∧ 𝑥 = 𝑦) ↔ ((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦))
5149, 50bitr2i 178 . . 3 (((𝑦 = ⟨ 𝑦, 𝐵⟩ ∧ 𝑦𝐴) ∧ 𝑥 = 𝑦) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴))
52 ancom 257 . . 3 ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥𝐴) ↔ (𝑥𝐴𝑦 = ⟨𝑥, 𝐵⟩))
5347, 51, 523bitri 199 . 2 ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = 𝑦) ↔ (𝑥𝐴𝑦 = ⟨𝑥, 𝐵⟩))
544, 1, 15, 17, 53en2i 6281 1 (𝐴 × {𝐵}) ≈ 𝐴
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   ↔ wb 102   = wceq 1259  ∃wex 1397   ∈ wcel 1409  Vcvv 2574  {csn 3403  ⟨cop 3406  ∩ cint 3643   class class class wbr 3792   × cxp 4371   ≈ cen 6250 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-en 6253 This theorem is referenced by:  xpsneng  6327  endisj  6329
 Copyright terms: Public domain W3C validator