Theorem xpcan2m 5044

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

Step	Hyp	Ref	Expression
1		ssxp1 5040	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ↔ 𝐴 ⊆ 𝐵))
2		ssxp1 5040	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → ((𝐵 × 𝐶) ⊆ (𝐴 × 𝐶) ↔ 𝐵 ⊆ 𝐴))
3	1, 2	anbi12d 465	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → (((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ⊆ (𝐴 × 𝐶)) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)))
4		eqss 3157	. 2 ⊢ ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ⊆ (𝐴 × 𝐶)))
5		eqss 3157	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
6	3, 4, 5	3bitr4g 222	1 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136   ⊆ wss 3116   × cxp 4602

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-dm 4614

This theorem is referenced by: (None)