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Mirrors > Home > MPE Home > Th. List > ackbij1lem4 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 10220. (Contributed by Stefan O'Rear, 19-Nov-2014.) |
Ref | Expression |
---|---|
ackbij1lem4 | ⊢ (𝐴 ∈ ω → {𝐴} ∈ (𝒫 ω ∩ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snelpwi 5436 | . 2 ⊢ (𝐴 ∈ ω → {𝐴} ∈ 𝒫 ω) | |
2 | snfi 9027 | . . 3 ⊢ {𝐴} ∈ Fin | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ ω → {𝐴} ∈ Fin) |
4 | 1, 3 | elind 4190 | 1 ⊢ (𝐴 ∈ ω → {𝐴} ∈ (𝒫 ω ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∩ cin 3943 𝒫 cpw 4596 {csn 4622 ωcom 7838 Fincfn 8922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-mo 2533 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-om 7839 df-1o 8448 df-en 8923 df-fin 8926 |
This theorem is referenced by: ackbij1lem8 10204 ackbij1lem14 10210 ackbij1lem16 10212 ackbij1lem18 10214 |
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