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Theorem xpcanm 4703
Description: Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
xpcanm (x x 𝐶 → ((𝐶 × A) = (𝐶 × B) ↔ A = B))
Distinct variable group:   x,𝐶
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem xpcanm
StepHypRef Expression
1 ssxp2 4701 . . 3 (x x 𝐶 → ((𝐶 × A) ⊆ (𝐶 × B) ↔ AB))
2 ssxp2 4701 . . 3 (x x 𝐶 → ((𝐶 × B) ⊆ (𝐶 × A) ↔ BA))
31, 2anbi12d 442 . 2 (x x 𝐶 → (((𝐶 × A) ⊆ (𝐶 × B) (𝐶 × B) ⊆ (𝐶 × A)) ↔ (AB BA)))
4 eqss 2954 . 2 ((𝐶 × A) = (𝐶 × B) ↔ ((𝐶 × A) ⊆ (𝐶 × B) (𝐶 × B) ⊆ (𝐶 × A)))
5 eqss 2954 . 2 (A = B ↔ (AB BA))
63, 4, 53bitr4g 212 1 (x x 𝐶 → ((𝐶 × A) = (𝐶 × B) ↔ A = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wss 2911   × cxp 4286
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by: (None)
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