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Theorem xpcan2m 4986
Description: Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
xpcan2m (∃𝑥 𝑥𝐶 → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem xpcan2m
StepHypRef Expression
1 ssxp1 4982 . . 3 (∃𝑥 𝑥𝐶 → ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ↔ 𝐴𝐵))
2 ssxp1 4982 . . 3 (∃𝑥 𝑥𝐶 → ((𝐵 × 𝐶) ⊆ (𝐴 × 𝐶) ↔ 𝐵𝐴))
31, 2anbi12d 465 . 2 (∃𝑥 𝑥𝐶 → (((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ⊆ (𝐴 × 𝐶)) ↔ (𝐴𝐵𝐵𝐴)))
4 eqss 3116 . 2 ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ⊆ (𝐴 × 𝐶)))
5 eqss 3116 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
63, 4, 53bitr4g 222 1 (∃𝑥 𝑥𝐶 → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wex 1469  wcel 1481  wss 3075   × cxp 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-dm 4556
This theorem is referenced by: (None)
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