Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpcan2m GIF version

Theorem xpcan2m 4761
 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 4757 . . 3 (∃𝑥 𝑥𝐶 → ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ↔ 𝐴𝐵))
2 ssxp1 4757 . . 3 (∃𝑥 𝑥𝐶 → ((𝐵 × 𝐶) ⊆ (𝐴 × 𝐶) ↔ 𝐵𝐴))
31, 2anbi12d 442 . 2 (∃𝑥 𝑥𝐶 → (((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ⊆ (𝐴 × 𝐶)) ↔ (𝐴𝐵𝐵𝐴)))
4 eqss 2960 . 2 ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ⊆ (𝐴 × 𝐶)))
5 eqss 2960 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
63, 4, 53bitr4g 212 1 (∃𝑥 𝑥𝐶 → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393   ⊆ wss 2917   × cxp 4343 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-dm 4355 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator