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| Mirrors > Home > MPE Home > Th. List > xpfir | Structured version Visualization version GIF version | ||
| Description: The components of a nonempty finite Cartesian product are finite. (Contributed by Paul Chapman, 11-Apr-2009.) (Proof shortened by Mario Carneiro, 29-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| xpfir | ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | xpexr2 7942 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
| 2 | 1 | simpld 494 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐴 ∈ V) | 
| 3 | 1 | simprd 495 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐵 ∈ V) | 
| 4 | simpr 484 | . . . . . 6 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 × 𝐵) ≠ ∅) | |
| 5 | xpnz 6178 | . . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | |
| 6 | 4, 5 | sylibr 234 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) | 
| 7 | 6 | simprd 495 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐵 ≠ ∅) | 
| 8 | xpdom3 9111 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵)) | |
| 9 | 2, 3, 7, 8 | syl3anc 1372 | . . 3 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵)) | 
| 10 | domfi 9230 | . . 3 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ 𝐴 ≼ (𝐴 × 𝐵)) → 𝐴 ∈ Fin) | |
| 11 | 9, 10 | syldan 591 | . 2 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐴 ∈ Fin) | 
| 12 | 6 | simpld 494 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐴 ≠ ∅) | 
| 13 | xpdom3 9111 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝐴 ≠ ∅) → 𝐵 ≼ (𝐵 × 𝐴)) | |
| 14 | 3, 2, 12, 13 | syl3anc 1372 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐵 ≼ (𝐵 × 𝐴)) | 
| 15 | xpcomeng 9105 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵)) | |
| 16 | 3, 2, 15 | syl2anc 584 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵)) | 
| 17 | domentr 9054 | . . . 4 ⊢ ((𝐵 ≼ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴 × 𝐵)) → 𝐵 ≼ (𝐴 × 𝐵)) | |
| 18 | 14, 16, 17 | syl2anc 584 | . . 3 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐵 ≼ (𝐴 × 𝐵)) | 
| 19 | domfi 9230 | . . 3 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝐴 × 𝐵)) → 𝐵 ∈ Fin) | |
| 20 | 18, 19 | syldan 591 | . 2 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐵 ∈ Fin) | 
| 21 | 11, 20 | jca 511 | 1 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∅c0 4332 class class class wbr 5142 × cxp 5682 ≈ cen 8983 ≼ cdom 8984 Fincfn 8986 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-om 7889 df-1st 8015 df-2nd 8016 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-fin 8990 | 
| This theorem is referenced by: hashxpe 32812 | 
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