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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 8387. (Contributed by FL, 3-Aug-2009.) |
Ref | Expression |
---|---|
unfir | ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 3927 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | ssfi 8340 | . . 3 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → 𝐴 ∈ Fin) | |
3 | 1, 2 | mpan2 671 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → 𝐴 ∈ Fin) |
4 | ssun2 3928 | . . 3 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
5 | ssfi 8340 | . . 3 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → 𝐵 ∈ Fin) | |
6 | 4, 5 | mpan2 671 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → 𝐵 ∈ Fin) |
7 | 3, 6 | jca 501 | 1 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 ∪ cun 3721 ⊆ wss 3723 Fincfn 8113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-om 7217 df-er 7900 df-en 8114 df-fin 8117 |
This theorem is referenced by: difinf 8390 hashunx 13377 eldioph4b 37899 |
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