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Theorem xpcan2 6133
Description: Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
Assertion
Ref Expression
xpcan2 (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))

Proof of Theorem xpcan2
StepHypRef Expression
1 xp11 6131 . . 3 ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐴 = 𝐵𝐶 = 𝐶)))
2 eqid 2733 . . . 4 𝐶 = 𝐶
32biantru 531 . . 3 (𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐶 = 𝐶))
41, 3bitr4di 289 . 2 ((𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
5 nne 2944 . . 3 𝐴 ≠ ∅ ↔ 𝐴 = ∅)
6 simpl 484 . . . . 5 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → 𝐴 = ∅)
7 xpeq1 5651 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 × 𝐶) = (∅ × 𝐶))
8 0xp 5734 . . . . . . . . . 10 (∅ × 𝐶) = ∅
97, 8eqtrdi 2789 . . . . . . . . 9 (𝐴 = ∅ → (𝐴 × 𝐶) = ∅)
109eqeq1d 2735 . . . . . . . 8 (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ∅ = (𝐵 × 𝐶)))
11 eqcom 2740 . . . . . . . 8 (∅ = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅)
1210, 11bitrdi 287 . . . . . . 7 (𝐴 = ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
1312adantr 482 . . . . . 6 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ (𝐵 × 𝐶) = ∅))
14 df-ne 2941 . . . . . . . 8 (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
15 xpeq0 6116 . . . . . . . . 9 ((𝐵 × 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅))
16 orel2 890 . . . . . . . . 9 𝐶 = ∅ → ((𝐵 = ∅ ∨ 𝐶 = ∅) → 𝐵 = ∅))
1715, 16biimtrid 241 . . . . . . . 8 𝐶 = ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
1814, 17sylbi 216 . . . . . . 7 (𝐶 ≠ ∅ → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
1918adantl 483 . . . . . 6 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐵 × 𝐶) = ∅ → 𝐵 = ∅))
2013, 19sylbid 239 . . . . 5 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐵 = ∅))
21 eqtr3 2759 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
226, 20, 21syl6an 683 . . . 4 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) → 𝐴 = 𝐵))
23 xpeq1 5651 . . . 4 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2422, 23impbid1 224 . . 3 ((𝐴 = ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
255, 24sylanb 582 . 2 ((¬ 𝐴 ≠ ∅ ∧ 𝐶 ≠ ∅) → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
264, 25pm2.61ian 811 1 (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wne 2940  c0 4286   × cxp 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648
This theorem is referenced by: (None)
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