Theorem xpexr 7625

Description: If a Cartesian product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)

Assertion

Ref	Expression
xpexr	⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V))

Proof of Theorem xpexr

Step	Hyp	Ref	Expression
1		0ex 5213	. . . . . 6 ⊢ ∅ ∈ V
2		eleq1 2902	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
3	1, 2	mpbiri 260	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
4	3	pm2.24d 154	. . . 4 ⊢ (𝐴 = ∅ → (¬ 𝐴 ∈ V → 𝐵 ∈ V))
5	4	a1d 25	. . 3 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)))
6		rnexg 7616	. . . . 5 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V)
7		rnxp 6029	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
8	7	eleq1d 2899	. . . . 5 ⊢ (𝐴 ≠ ∅ → (ran (𝐴 × 𝐵) ∈ V ↔ 𝐵 ∈ V))
9	6, 8	syl5ib 246	. . . 4 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → 𝐵 ∈ V))
10	9	a1dd 50	. . 3 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)))
11	5, 10	pm2.61ine 3102	. 2 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V))
12	11	orrd 859	1 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  Vcvv 3496  ∅c0 4293   × cxp 5555  ran crn 5558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568

This theorem is referenced by: (None)