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Mirrors > Home > MPE Home > Th. List > nnfi | Structured version Visualization version GIF version |
Description: Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
nnfi | ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onfin2 8440 | . . 3 ⊢ ω = (On ∩ Fin) | |
2 | inss2 4053 | . . 3 ⊢ (On ∩ Fin) ⊆ Fin | |
3 | 1, 2 | eqsstri 3853 | . 2 ⊢ ω ⊆ Fin |
4 | 3 | sseli 3816 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∩ cin 3790 Oncon0 5976 ωcom 7343 Fincfn 8241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-om 7344 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 |
This theorem is referenced by: cardnn 9122 en2eqpr 9163 en2eleq 9164 infxpenlem 9169 dfac12k 9304 pwsdompw 9361 ackbij2lem1 9376 ackbij1lem3 9379 ackbij1lem5 9381 ackbij1lem14 9390 ackbij1b 9396 fin23lem23 9483 fin23lem22 9484 domtriomlem 9599 gchcda1 9813 gch2 9832 omina 9848 hashgval2 13482 hashdom 13483 hashp1i 13505 hash1snb 13521 hash2pr 13565 pr2pwpr 13575 hash3tr 13586 xpsfrnel 16609 symggen 18273 psgnunilem1 18296 lt6abl 18682 znfld 20304 frgpcyg 20317 xpsmet 22595 xpsxms 22747 xpsms 22748 isppw 25292 finxpreclem4 33826 harinf 38542 frlmpwfi 38609 |
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