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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 7927 | . . . . 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 6163 | . . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | |
6 | 4, 5 | sylibr 233 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
7 | 6 | simprd 495 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐵 ≠ ∅) |
8 | xpdom3 9094 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵)) | |
9 | 2, 3, 7, 8 | syl3anc 1369 | . . 3 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵)) |
10 | domfi 9216 | . . 3 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ 𝐴 ≼ (𝐴 × 𝐵)) → 𝐴 ∈ Fin) | |
11 | 9, 10 | syldan 590 | . 2 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐴 ∈ Fin) |
12 | 6 | simpld 494 | . . . . 5 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐴 ≠ ∅) |
13 | xpdom3 9094 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝐴 ≠ ∅) → 𝐵 ≼ (𝐵 × 𝐴)) | |
14 | 3, 2, 12, 13 | syl3anc 1369 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐵 ≼ (𝐵 × 𝐴)) |
15 | xpcomeng 9088 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵)) | |
16 | 3, 2, 15 | syl2anc 583 | . . . 4 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵)) |
17 | domentr 9033 | . . . 4 ⊢ ((𝐵 ≼ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴 × 𝐵)) → 𝐵 ≼ (𝐴 × 𝐵)) | |
18 | 14, 16, 17 | syl2anc 583 | . . 3 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → 𝐵 ≼ (𝐴 × 𝐵)) |
19 | domfi 9216 | . . 3 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝐴 × 𝐵)) → 𝐵 ∈ Fin) | |
20 | 18, 19 | syldan 590 | . 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 2099 ≠ wne 2937 Vcvv 3471 ∅c0 4323 class class class wbr 5148 × cxp 5676 ≈ cen 8960 ≼ cdom 8961 Fincfn 8963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-om 7871 df-1st 7993 df-2nd 7994 df-1o 8486 df-er 8724 df-en 8964 df-dom 8965 df-fin 8967 |
This theorem is referenced by: hashxpe 32576 |
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