Theorem xpcanm 5131

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

Step	Hyp	Ref	Expression
1		ssxp2 5129	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) ↔ 𝐴 ⊆ 𝐵))
2		ssxp2 5129	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → ((𝐶 × 𝐵) ⊆ (𝐶 × 𝐴) ↔ 𝐵 ⊆ 𝐴))
3	1, 2	anbi12d 473	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → (((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) ∧ (𝐶 × 𝐵) ⊆ (𝐶 × 𝐴)) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)))
4		eqss 3212	. 2 ⊢ ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) ∧ (𝐶 × 𝐵) ⊆ (𝐶 × 𝐴)))
5		eqss 3212	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
6	3, 4, 5	3bitr4g 223	1 ⊢ (∃𝑥 𝑥 ∈ 𝐶 → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1516   ∈ wcel 2177   ⊆ wss 3170   × cxp 4681

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-xp 4689  df-rel 4690  df-cnv 4691  df-dm 4693  df-rn 4694

This theorem is referenced by: (None)