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Theorem xpcan 5605
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 5604 . . 3 ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 = 𝐶𝐴 = 𝐵)))
2 eqid 2651 . . . 4 𝐶 = 𝐶
32biantrur 526 . . 3 (𝐴 = 𝐵 ↔ (𝐶 = 𝐶𝐴 = 𝐵))
41, 3syl6bbr 278 . 2 ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
5 nne 2827 . . . 4 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6 simpr 476 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → 𝐴 = ∅)
7 xpeq2 5163 . . . . . . . . . 10 (𝐴 = ∅ → (𝐶 × 𝐴) = (𝐶 × ∅))
8 xp0 5587 . . . . . . . . . 10 (𝐶 × ∅) = ∅
97, 8syl6eq 2701 . . . . . . . . 9 (𝐴 = ∅ → (𝐶 × 𝐴) = ∅)
109eqeq1d 2653 . . . . . . . 8 (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ∅ = (𝐶 × 𝐵)))
11 eqcom 2658 . . . . . . . 8 (∅ = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅)
1210, 11syl6bb 276 . . . . . . 7 (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
1312adantl 481 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
14 df-ne 2824 . . . . . . . 8 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15 xpeq0 5589 . . . . . . . . 9 ((𝐶 × 𝐵) = ∅ ↔ (𝐶 = ∅ ∨ 𝐵 = ∅))
16 orel1 396 . . . . . . . . 9 𝐶 = ∅ → ((𝐶 = ∅ ∨ 𝐵 = ∅) → 𝐵 = ∅))
1715, 16syl5bi 232 . . . . . . . 8 𝐶 = ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
1814, 17sylbi 207 . . . . . . 7 (𝐶 ≠ ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
1918adantr 480 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
2013, 19sylbid 230 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐵 = ∅))
21 eqtr3 2672 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
226, 20, 21syl6an 567 . . . 4 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
235, 22sylan2b 491 . . 3 ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
24 xpeq2 5163 . . 3 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
2523, 24impbid1 215 . 2 ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
264, 25pm2.61dan 849 1 (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wne 2823  c0 3948   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by: (None)
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