xpexr2 - Metamath Proof Explorer

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Theorem xpexr2 7107

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 5553	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
2		dmxp 5344	. . . . . 6 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
3	2	adantl 482	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) = 𝐴)
4		dmexg 7097	. . . . . 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 2702	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
7		rnxp 5564	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
8	7	adantl 482	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → ran (𝐴 × 𝐵) = 𝐵)
9		rnexg 7098	. . . . . 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 2702	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ V)
12	6, 11	anim12dan 882	. . 3 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
13	12	ancom2s 844	. 2 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14	1, 13	sylan2br 493	1 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∅c0 3915 × cxp 5112 dom cdm 5114 ran crn 5115

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125

This theorem is referenced by: xpfir 8182 bj-xpnzex 32946