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Mirrors > Home > MPE Home > Th. List > ackbij1lem4 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 9743. (Contributed by Stefan O'Rear, 19-Nov-2014.) |
Ref | Expression |
---|---|
ackbij1lem4 | ⊢ (𝐴 ∈ ω → {𝐴} ∈ (𝒫 ω ∩ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snelpwi 5303 | . 2 ⊢ (𝐴 ∈ ω → {𝐴} ∈ 𝒫 ω) | |
2 | snfi 8642 | . . 3 ⊢ {𝐴} ∈ Fin | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ ω → {𝐴} ∈ Fin) |
4 | 1, 3 | elind 4084 | 1 ⊢ (𝐴 ∈ ω → {𝐴} ∈ (𝒫 ω ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∩ cin 3842 𝒫 cpw 4488 {csn 4516 ωcom 7599 Fincfn 8555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-om 7600 df-1o 8131 df-en 8556 df-fin 8559 |
This theorem is referenced by: ackbij1lem8 9727 ackbij1lem14 9733 ackbij1lem16 9735 ackbij1lem18 9737 |
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