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Mirrors > Home > MPE Home > Th. List > ackbij1lem11 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 9835. (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 5205 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ (𝒫 ω ∩ Fin)) → 𝐵 ∈ V) | |
2 | elinel1 4099 | . . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω) | |
3 | 2 | elpwid 4514 | . . . . 5 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω) |
4 | sstr 3899 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ ω) → 𝐵 ⊆ ω) | |
5 | 3, 4 | sylan2 596 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ (𝒫 ω ∩ Fin)) → 𝐵 ⊆ ω) |
6 | 1, 5 | elpwd 4511 | . . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ (𝒫 ω ∩ Fin)) → 𝐵 ∈ 𝒫 ω) |
7 | 6 | ancoms 462 | . 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝒫 ω) |
8 | elinel2 4100 | . . 3 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin) | |
9 | ssfi 8840 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | |
10 | 8, 9 | sylan 583 | . 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) |
11 | 7, 10 | elind 4098 | 1 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (𝒫 ω ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ∩ cin 3856 ⊆ wss 3857 𝒫 cpw 4503 {csn 4531 ∪ ciun 4894 ↦ cmpt 5124 × cxp 5538 ‘cfv 6369 ωcom 7633 Fincfn 8615 cardccrd 9534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-om 7634 df-1o 8191 df-en 8616 df-fin 8619 |
This theorem is referenced by: ackbij1lem12 9828 ackbij1lem15 9831 ackbij1lem16 9832 ackbij1lem18 9834 |
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