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Theorem xpcan 6165
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 6164 . . 3 ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 = 𝐶𝐴 = 𝐵)))
2 eqid 2724 . . . 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 2936 . . . 4 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6 simpr 484 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → 𝐴 = ∅)
7 xpeq2 5687 . . . . . . . . . 10 (𝐴 = ∅ → (𝐶 × 𝐴) = (𝐶 × ∅))
8 xp0 6147 . . . . . . . . . 10 (𝐶 × ∅) = ∅
97, 8eqtrdi 2780 . . . . . . . . 9 (𝐴 = ∅ → (𝐶 × 𝐴) = ∅)
109eqeq1d 2726 . . . . . . . 8 (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ∅ = (𝐶 × 𝐵)))
11 eqcom 2731 . . . . . . . 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 2933 . . . . . . . 8 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15 xpeq0 6149 . . . . . . . . 9 ((𝐶 × 𝐵) = ∅ ↔ (𝐶 = ∅ ∨ 𝐵 = ∅))
16 orel1 885 . . . . . . . . 9 𝐶 = ∅ → ((𝐶 = ∅ ∨ 𝐵 = ∅) → 𝐵 = ∅))
1715, 16biimtrid 241 . . . . . . . 8 𝐶 = ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
1814, 17sylbi 216 . . . . . . 7 (𝐶 ≠ ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
1918adantr 480 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
2013, 19sylbid 239 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐵 = ∅))
21 eqtr3 2750 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
226, 20, 21syl6an 681 . . . 4 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
235, 22sylan2b 593 . . 3 ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
24 xpeq2 5687 . . 3 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
2523, 24impbid1 224 . 2 ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
264, 25pm2.61dan 810 1 (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844   = wceq 1533  wne 2932  c0 4314   × cxp 5664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677
This theorem is referenced by: (None)
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