Theorem xpcan2m 5110

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 5106	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ↔ 𝐴 ⊆ 𝐵))
2		ssxp1 5106	. . 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 3198	. 2 ⊢ ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ⊆ (𝐴 × 𝐶)))
5		eqss 3198	. 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 1364  ∃wex 1506   ∈ wcel 2167   ⊆ wss 3157   × cxp 4661

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-dm 4673

This theorem is referenced by: (None)