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Mirrors > Home > MPE Home > Th. List > ackbij1lem4 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 9999. (Contributed by Stefan O'Rear, 19-Nov-2014.) |
Ref | Expression |
---|---|
ackbij1lem4 | ⊢ (𝐴 ∈ ω → {𝐴} ∈ (𝒫 ω ∩ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snelpwi 5360 | . 2 ⊢ (𝐴 ∈ ω → {𝐴} ∈ 𝒫 ω) | |
2 | snfi 8834 | . . 3 ⊢ {𝐴} ∈ Fin | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ ω → {𝐴} ∈ Fin) |
4 | 1, 3 | elind 4128 | 1 ⊢ (𝐴 ∈ ω → {𝐴} ∈ (𝒫 ω ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∩ cin 3886 𝒫 cpw 4533 {csn 4561 ωcom 7712 Fincfn 8733 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-om 7713 df-1o 8297 df-en 8734 df-fin 8737 |
This theorem is referenced by: ackbij1lem8 9983 ackbij1lem14 9989 ackbij1lem16 9991 ackbij1lem18 9993 |
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