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| Description: Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 18-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| ackbij2lem1 | ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ordom 7897 | . . . . . . 7 ⊢ Ord ω | |
| 2 | ordelss 6400 | . . . . . . 7 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω) | |
| 3 | 1, 2 | mpan 690 | . . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) | 
| 4 | 3 | sspwd 4613 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ 𝒫 ω) | 
| 5 | 4 | sselda 3983 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → 𝑎 ∈ 𝒫 ω) | 
| 6 | nnfi 9207 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
| 7 | elpwi 4607 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴) | |
| 8 | ssfi 9213 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑎 ⊆ 𝐴) → 𝑎 ∈ Fin) | |
| 9 | 6, 7, 8 | syl2an 596 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → 𝑎 ∈ Fin) | 
| 10 | 5, 9 | elind 4200 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → 𝑎 ∈ (𝒫 ω ∩ Fin)) | 
| 11 | 10 | ex 412 | . 2 ⊢ (𝐴 ∈ ω → (𝑎 ∈ 𝒫 𝐴 → 𝑎 ∈ (𝒫 ω ∩ Fin))) | 
| 12 | 11 | ssrdv 3989 | 1 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 Ord word 6383 ωcom 7887 Fincfn 8985 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-en 8986 df-fin 8989 | 
| This theorem is referenced by: ackbij1b 10278 ackbij2lem2 10279 | 
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