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Theorem xpcan 5812
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 5811 . . 3 ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 = 𝐶𝐴 = 𝐵)))
2 eqid 2826 . . . 4 𝐶 = 𝐶
32biantrur 528 . . 3 (𝐴 = 𝐵 ↔ (𝐶 = 𝐶𝐴 = 𝐵))
41, 3syl6bbr 281 . 2 ((𝐶 ≠ ∅ ∧ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
5 nne 3004 . . . 4 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6 simpr 479 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → 𝐴 = ∅)
7 xpeq2 5364 . . . . . . . . . 10 (𝐴 = ∅ → (𝐶 × 𝐴) = (𝐶 × ∅))
8 xp0 5794 . . . . . . . . . 10 (𝐶 × ∅) = ∅
97, 8syl6eq 2878 . . . . . . . . 9 (𝐴 = ∅ → (𝐶 × 𝐴) = ∅)
109eqeq1d 2828 . . . . . . . 8 (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ∅ = (𝐶 × 𝐵)))
11 eqcom 2833 . . . . . . . 8 (∅ = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅)
1210, 11syl6bb 279 . . . . . . 7 (𝐴 = ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
1312adantl 475 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ (𝐶 × 𝐵) = ∅))
14 df-ne 3001 . . . . . . . 8 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15 xpeq0 5796 . . . . . . . . 9 ((𝐶 × 𝐵) = ∅ ↔ (𝐶 = ∅ ∨ 𝐵 = ∅))
16 orel1 919 . . . . . . . . 9 𝐶 = ∅ → ((𝐶 = ∅ ∨ 𝐵 = ∅) → 𝐵 = ∅))
1715, 16syl5bi 234 . . . . . . . 8 𝐶 = ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
1814, 17sylbi 209 . . . . . . 7 (𝐶 ≠ ∅ → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
1918adantr 474 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐵) = ∅ → 𝐵 = ∅))
2013, 19sylbid 232 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐵 = ∅))
21 eqtr3 2849 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
226, 20, 21syl6an 676 . . . 4 ((𝐶 ≠ ∅ ∧ 𝐴 = ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
235, 22sylan2b 589 . . 3 ((𝐶 ≠ ∅ ∧ ¬ 𝐴 ≠ ∅) → ((𝐶 × 𝐴) = (𝐶 × 𝐵) → 𝐴 = 𝐵))
24 xpeq2 5364 . . 3 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
2523, 24impbid1 217 . 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 198  wa 386  wo 880   = wceq 1658  wne 3000  c0 4145   × cxp 5341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-xp 5349  df-rel 5350  df-cnv 5351  df-dm 5353  df-rn 5354
This theorem is referenced by: (None)
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