Theorem xpcan2m 5202

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 5198	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ↔ 𝐴 ⊆ 𝐵))
2		ssxp1 5198	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → ((𝐵 × 𝐶) ⊆ (𝐴 × 𝐶) ↔ 𝐵 ⊆ 𝐴))
3	1, 2	anbi12d 473	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → (((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ⊆ (𝐴 × 𝐶)) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)))
4		eqss 3252	. 2 ⊢ ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ⊆ (𝐴 × 𝐶)))
5		eqss 3252	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
6	3, 4, 5	3bitr4g 223	1 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2203   ⊆ wss 3210   × cxp 4746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-xp 4754  df-dm 4758

This theorem is referenced by: (None)