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Theorem xpcan2m 4688
Description: Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
xpcan2m (x x 𝐶 → ((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 4684 . . 3 (x x 𝐶 → ((A × 𝐶) ⊆ (B × 𝐶) ↔ AB))
2 ssxp1 4684 . . 3 (x x 𝐶 → ((B × 𝐶) ⊆ (A × 𝐶) ↔ BA))
31, 2anbi12d 445 . 2 (x x 𝐶 → (((A × 𝐶) ⊆ (B × 𝐶) (B × 𝐶) ⊆ (A × 𝐶)) ↔ (AB BA)))
4 eqss 2937 . 2 ((A × 𝐶) = (B × 𝐶) ↔ ((A × 𝐶) ⊆ (B × 𝐶) (B × 𝐶) ⊆ (A × 𝐶)))
5 eqss 2937 . 2 (A = B ↔ (AB BA))
63, 4, 53bitr4g 212 1 (x x 𝐶 → ((A × 𝐶) = (B × 𝐶) ↔ A = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  wss 2894   × cxp 4270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-dm 4282
This theorem is referenced by: (None)
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