ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpcanm Structured version   GIF version

Theorem xpcanm 4683
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 4681 . . 3 (x x 𝐶 → ((𝐶 × A) ⊆ (𝐶 × B) ↔ AB))
2 ssxp2 4681 . . 3 (x x 𝐶 → ((𝐶 × B) ⊆ (𝐶 × A) ↔ BA))
31, 2anbi12d 445 . 2 (x x 𝐶 → (((𝐶 × A) ⊆ (𝐶 × B) (𝐶 × B) ⊆ (𝐶 × A)) ↔ (AB BA)))
4 eqss 2933 . 2 ((𝐶 × A) = (𝐶 × B) ↔ ((𝐶 × A) ⊆ (𝐶 × B) (𝐶 × B) ⊆ (𝐶 × A)))
5 eqss 2933 . 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 1226  wex 1358   wcel 1370  wss 2890   × cxp 4266
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276  df-dm 4278  df-rn 4279
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator