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Mirrors > Home > MPE Home > Th. List > unfir | Structured version Visualization version GIF version |
Description: If a union is finite, the operands are finite. Converse of unfi 9190. (Contributed by FL, 3-Aug-2009.) |
Ref | Expression |
---|---|
unfir | ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4168 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | ssfi 9191 | . . 3 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → 𝐴 ∈ Fin) | |
3 | 1, 2 | mpan2 690 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → 𝐴 ∈ Fin) |
4 | ssun2 4169 | . . 3 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
5 | ssfi 9191 | . . 3 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → 𝐵 ∈ Fin) | |
6 | 4, 5 | mpan2 690 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → 𝐵 ∈ Fin) |
7 | 3, 6 | jca 511 | 1 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ∪ cun 3943 ⊆ wss 3945 Fincfn 8957 |
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 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
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 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7865 df-1o 8480 df-en 8958 df-fin 8961 |
This theorem is referenced by: difinf 9334 hashunx 14371 eldioph4b 42225 |
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