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Mirrors > Home > MPE Home > Th. List > ackbij1lem11 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 10274. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
Ref | Expression |
---|---|
ackbij.f | ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) |
Ref | Expression |
---|---|
ackbij1lem11 | ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (𝒫 ω ∩ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 5328 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ (𝒫 ω ∩ Fin)) → 𝐵 ∈ V) | |
2 | elinel1 4210 | . . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω) | |
3 | 2 | elpwid 4613 | . . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω) |
4 | sstr 4003 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ ω) → 𝐵 ⊆ ω) | |
5 | 3, 4 | sylan2 593 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ (𝒫 ω ∩ Fin)) → 𝐵 ⊆ ω) |
6 | 1, 5 | elpwd 4610 | . . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ (𝒫 ω ∩ Fin)) → 𝐵 ∈ 𝒫 ω) |
7 | 6 | ancoms 458 | . 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝒫 ω) |
8 | elinel2 4211 | . . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin) | |
9 | ssfi 9211 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | |
10 | 8, 9 | sylan 580 | . 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) |
11 | 7, 10 | elind 4209 | 1 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (𝒫 ω ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 𝒫 cpw 4604 {csn 4630 ∪ ciun 4995 ↦ cmpt 5230 × cxp 5686 ‘cfv 6562 ωcom 7886 Fincfn 8983 cardccrd 9972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-om 7887 df-1o 8504 df-en 8984 df-fin 8987 |
This theorem is referenced by: ackbij1lem12 10267 ackbij1lem15 10270 ackbij1lem16 10271 ackbij1lem18 10273 |
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