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Mirrors > Home > MPE Home > Th. List > ackbij1lem6 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 8948. (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 | inss1 3795 | . . . . 5 ⊢ (𝒫 ω ∩ Fin) ⊆ 𝒫 ω | |
2 | 1 | sseli 3564 | . . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω) |
3 | 1 | sseli 3564 | . . . 4 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω) |
4 | elpwi 4117 | . . . . 5 ⊢ (𝐴 ∈ 𝒫 ω → 𝐴 ⊆ ω) | |
5 | elpwi 4117 | . . . . 5 ⊢ (𝐵 ∈ 𝒫 ω → 𝐵 ⊆ ω) | |
6 | simpl 472 | . . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ 𝐵 ⊆ ω) → 𝐴 ⊆ ω) | |
7 | simpr 476 | . . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ 𝐵 ⊆ ω) → 𝐵 ⊆ ω) | |
8 | 6, 7 | unssd 3751 | . . . . 5 ⊢ ((𝐴 ⊆ ω ∧ 𝐵 ⊆ ω) → (𝐴 ∪ 𝐵) ⊆ ω) |
9 | 4, 5, 8 | syl2an 493 | . . . 4 ⊢ ((𝐴 ∈ 𝒫 ω ∧ 𝐵 ∈ 𝒫 ω) → (𝐴 ∪ 𝐵) ⊆ ω) |
10 | 2, 3, 9 | syl2an 493 | . . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ⊆ ω) |
11 | inss2 3796 | . . . . . 6 ⊢ (𝒫 ω ∩ Fin) ⊆ Fin | |
12 | 11 | sseli 3564 | . . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin) |
13 | 11 | sseli 3564 | . . . . 5 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin) |
14 | unfi 8112 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
15 | 12, 13, 14 | syl2an 493 | . . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ Fin) |
16 | elpwg 4116 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → ((𝐴 ∪ 𝐵) ∈ 𝒫 ω ↔ (𝐴 ∪ 𝐵) ⊆ ω)) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐴 ∪ 𝐵) ∈ 𝒫 ω ↔ (𝐴 ∪ 𝐵) ⊆ ω)) |
18 | 10, 17 | mpbird 246 | . 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ 𝒫 ω) |
19 | 18, 15 | elind 3760 | 1 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ωcom 6957 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-er 7629 df-en 7842 df-fin 7845 |
This theorem is referenced by: ackbij1lem9 8933 ackbij1lem18 8942 |
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