xpexr2 - Metamath Proof Explorer

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Theorem xpexr2 7866

Description: If a nonempty Cartesian product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.)

**Assertion**
Ref	Expression
xpexr2	⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

**Proof of Theorem xpexr2**
Step	Hyp	Ref	Expression
1		xpnz 6117	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
2		dmxp 5878	. . . . . 6 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
3	2	adantl 482	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) = 𝐴)
4		dmexg 7848	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → dom (𝐴 × 𝐵) ∈ V)
5	4	adantr 481	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) ∈ V)
6	3, 5	eqeltrrd 2841	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
7		rnxp 6128	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
8	7	adantl 482	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → ran (𝐴 × 𝐵) = 𝐵)
9		rnexg 7849	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V)
10	9	adantr 481	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → ran (𝐴 × 𝐵) ∈ V)
11	8, 10	eqeltrrd 2841	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ V)
12	6, 11	anim12dan 625	. . 3 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
13	12	ancom2s 656	. 2 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14	1, 13	sylan2br 601	1 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 Vcvv 3432 ∅c0 4268 × cxp 5623 dom cdm 5625 ran crn 5626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-11 2168 ax-ext 2712 ax-sep 5225 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-sb 2074 df-clab 2719 df-cleq 2732 df-clel 2815 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-xp 5631 df-rel 5632 df-cnv 5633 df-dm 5635 df-rn 5636

This theorem is referenced by: xpfir 9175 bj-xpnzex 37319