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Mirrors > Home > MPE Home > Th. List > ackbij1lem6 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 9653. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
Ref | Expression |
---|---|
ackbij1lem6 | ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinel2 4170 | . . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin) | |
2 | elinel2 4170 | . . . 4 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin) | |
3 | unfi 8773 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
4 | 1, 2, 3 | syl2an 595 | . . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ Fin) |
5 | elinel1 4169 | . . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω) | |
6 | elinel1 4169 | . . . 4 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω) | |
7 | elpwi 4547 | . . . . 5 ⊢ (𝐴 ∈ 𝒫 ω → 𝐴 ⊆ ω) | |
8 | elpwi 4547 | . . . . 5 ⊢ (𝐵 ∈ 𝒫 ω → 𝐵 ⊆ ω) | |
9 | simpl 483 | . . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ 𝐵 ⊆ ω) → 𝐴 ⊆ ω) | |
10 | simpr 485 | . . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ 𝐵 ⊆ ω) → 𝐵 ⊆ ω) | |
11 | 9, 10 | unssd 4159 | . . . . 5 ⊢ ((𝐴 ⊆ ω ∧ 𝐵 ⊆ ω) → (𝐴 ∪ 𝐵) ⊆ ω) |
12 | 7, 8, 11 | syl2an 595 | . . . 4 ⊢ ((𝐴 ∈ 𝒫 ω ∧ 𝐵 ∈ 𝒫 ω) → (𝐴 ∪ 𝐵) ⊆ ω) |
13 | 5, 6, 12 | syl2an 595 | . . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ⊆ ω) |
14 | 4, 13 | elpwd 4546 | . 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ 𝒫 ω) |
15 | 14, 4 | elind 4168 | 1 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ∪ cun 3931 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 ωcom 7569 Fincfn 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-oadd 8095 df-er 8278 df-en 8498 df-fin 8501 |
This theorem is referenced by: ackbij1lem9 9638 ackbij1lem18 9647 |
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