![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5983 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | |
2 | dmxp 5763 | . . . . . 6 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
3 | 2 | adantl 485 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) = 𝐴) |
4 | dmexg 7594 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → dom (𝐴 × 𝐵) ∈ V) | |
5 | 4 | adantr 484 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) ∈ V) |
6 | 3, 5 | eqeltrrd 2891 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V) |
7 | rnxp 5994 | . . . . . 6 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) | |
8 | 7 | adantl 485 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → ran (𝐴 × 𝐵) = 𝐵) |
9 | rnexg 7595 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V) | |
10 | 9 | adantr 484 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → ran (𝐴 × 𝐵) ∈ V) |
11 | 8, 10 | eqeltrrd 2891 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ V) |
12 | 6, 11 | anim12dan 621 | . . 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 597 | 1 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 × cxp 5517 dom cdm 5519 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: xpfir 8724 bj-xpnzex 34395 |
Copyright terms: Public domain | W3C validator |