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Mirrors > Home > MPE Home > Th. List > xpexr2 | Structured version Visualization version GIF version |
Description: If a nonempty Cartesian product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.) |
Ref | Expression |
---|---|
xpexr2 | ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpnz 6155 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | |
2 | dmxp 5926 | . . . . . 6 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
3 | 2 | adantl 483 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) = 𝐴) |
4 | dmexg 7889 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → dom (𝐴 × 𝐵) ∈ V) | |
5 | 4 | adantr 482 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) ∈ V) |
6 | 3, 5 | eqeltrrd 2835 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V) |
7 | rnxp 6166 | . . . . . 6 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) | |
8 | 7 | adantl 483 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → ran (𝐴 × 𝐵) = 𝐵) |
9 | rnexg 7890 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V) | |
10 | 9 | adantr 482 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → ran (𝐴 × 𝐵) ∈ V) |
11 | 8, 10 | eqeltrrd 2835 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ V) |
12 | 6, 11 | anim12dan 620 | . . 3 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅)) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
13 | 12 | ancom2s 649 | . 2 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
14 | 1, 13 | sylan2br 596 | 1 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∅c0 4321 × cxp 5673 dom cdm 5675 ran crn 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-dm 5685 df-rn 5686 |
This theorem is referenced by: xpfir 9262 bj-xpnzex 35778 |
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