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Mirrors > Home > MPE Home > Th. List > ackbij2lem1 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 10187. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
Ref | Expression |
---|---|
ackbij2lem1 | ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 7816 | . . . . . . 7 ⊢ Ord ω | |
2 | ordelss 6337 | . . . . . . 7 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω) | |
3 | 1, 2 | mpan 689 | . . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) |
4 | 3 | sspwd 4577 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ 𝒫 ω) |
5 | 4 | sselda 3948 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → 𝑎 ∈ 𝒫 ω) |
6 | nnfi 9117 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
7 | elpwi 4571 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴) | |
8 | ssfi 9123 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑎 ⊆ 𝐴) → 𝑎 ∈ Fin) | |
9 | 6, 7, 8 | syl2an 597 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → 𝑎 ∈ Fin) |
10 | 5, 9 | elind 4158 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → 𝑎 ∈ (𝒫 ω ∩ Fin)) |
11 | 10 | ex 414 | . 2 ⊢ (𝐴 ∈ ω → (𝑎 ∈ 𝒫 𝐴 → 𝑎 ∈ (𝒫 ω ∩ Fin))) |
12 | 11 | ssrdv 3954 | 1 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ∩ cin 3913 ⊆ wss 3914 𝒫 cpw 4564 Ord word 6320 ωcom 7806 Fincfn 8889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-om 7807 df-1o 8416 df-en 8890 df-fin 8893 |
This theorem is referenced by: ackbij1b 10183 ackbij2lem2 10184 |
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