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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.) Avoid ax-pow 5365. (Revised by BTernaryTau, 23-Sep-2024.) |
| Ref | Expression |
|---|---|
| nnfi | ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | enrefnn 9087 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) | |
| 2 | breq2 5147 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴)) | |
| 3 | 2 | rspcev 3622 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≈ 𝐴) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
| 4 | 1, 3 | mpdan 687 | . 2 ⊢ (𝐴 ∈ ω → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
| 5 | isfi 9016 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
| 6 | 4, 5 | sylibr 234 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 ωcom 7887 ≈ cen 8982 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-om 7888 df-en 8986 df-fin 8989 |
| This theorem is referenced by: ssnnfi 9209 enfii 9226 phplem1 9244 phplem2 9245 php 9247 php2 9248 php3 9249 nndomog 9253 onomeneq 9265 sucdom 9271 ominf 9294 findcard3 9318 nnsdomg 9335 infsdomnn 9338 fiint 9366 cardnn 10003 en2eqpr 10047 en2eleq 10048 infxpenlem 10053 dfac12k 10188 ficardadju 10240 pwsdompw 10243 ackbij2lem1 10258 ackbij1lem3 10261 ackbij1lem5 10263 ackbij1lem14 10272 ackbij1b 10278 fin23lem23 10366 fin23lem22 10367 domtriomlem 10482 gchdju1 10696 gch2 10715 omina 10731 hashgval2 14417 hashdom 14418 hashp1i 14442 hash1snb 14458 hash2pr 14508 pr2pwpr 14518 hash3tr 14530 xpsfrnel 17607 symggen 19488 psgnunilem1 19511 lt6abl 19913 simpgnsgd 20120 znfld 21579 frgpcyg 21592 xpsmet 24392 xpsxms 24547 xpsms 24548 isppw 27157 madefi 27950 oldfi 27951 unidifsnel 32553 unidifsnne 32554 finxpreclem4 37395 harinf 43046 frlmpwfi 43110 cantnfub2 43335 infordmin 43545 |
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