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Theorem xpcan 6196
Description: Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
Assertion
Ref Expression
xpcan (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))

Proof of Theorem xpcan
StepHypRef Expression
1 xp11 6195 . . 3 ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 = 𝐶𝐴 = 𝐵)))
2 eqid 2737 . . . 4 𝐶 = 𝐶
32biantrur 530 . . 3 (𝐴 = 𝐵 ↔ (𝐶 = 𝐶𝐴 = 𝐵))
41, 3bitr4di 289 . 2 ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
5 nne 2944 . . . 4 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6 simpr 484 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → 𝐴 = ∅)
7 xpeq2 5706 . . . . . . . . . 10 (𝐴 = ∅ → (𝐶 × 𝐴) = (𝐶 × ∅))
8 xp0 6178 . . . . . . . . . 10 (𝐶 × ∅) = ∅
97, 8eqtrdi 2793 . . . . . . . . 9 (𝐴 = ∅ → (𝐶 × 𝐴) = ∅)
109eqeq1d 2739 . . . . . . . 8 (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ∅ = (𝐶 × 𝐵)))
11 eqcom 2744 . . . . . . . 8 (∅ = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅)
1210, 11bitrdi 287 . . . . . . 7 (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
1312adantl 481 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
14 df-ne 2941 . . . . . . . 8 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15 xpeq0 6180 . . . . . . . . 9 ((𝐶 × 𝐵) = ∅ ↔ (𝐶 = ∅ ∨ 𝐵 = ∅))
16 orel1 889 . . . . . . . . 9 𝐶 = ∅ → ((𝐶 = ∅ ∨ 𝐵 = ∅) → 𝐵 = ∅))
1715, 16biimtrid 242 . . . . . . . 8 𝐶 = ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
1814, 17sylbi 217 . . . . . . 7 (𝐶 ≠ ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
1918adantr 480 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
2013, 19sylbid 240 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐵 = ∅))
21 eqtr3 2763 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
226, 20, 21syl6an 684 . . . 4 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
235, 22sylan2b 594 . . 3 ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
24 xpeq2 5706 . . 3 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
2523, 24impbid1 225 . 2 ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
264, 25pm2.61dan 813 1 (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wne 2940  c0 4333   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by: (None)
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