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Mirrors > Home > MPE Home > Th. List > ackbij2lem1 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 10280. (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 6402 | . . . . . . 7 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω) | |
3 | 1, 2 | mpan 690 | . . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) |
4 | 3 | sspwd 4618 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ 𝒫 ω) |
5 | 4 | sselda 3995 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → 𝑎 ∈ 𝒫 ω) |
6 | nnfi 9206 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
7 | elpwi 4612 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴) | |
8 | ssfi 9212 | . . . . 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 4210 | . . 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 4001 | 1 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 Ord word 6385 ωcom 7887 Fincfn 8984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-en 8985 df-fin 8988 |
This theorem is referenced by: ackbij1b 10276 ackbij2lem2 10277 |
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