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Mirrors > Home > ILE Home > Th. List > nnfi | 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 | enrefg 6778 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) | |
2 | breq2 4019 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴)) | |
3 | 2 | rspcev 2853 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≈ 𝐴) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
4 | 1, 3 | mpdan 421 | . 2 ⊢ (𝐴 ∈ ω → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
5 | isfi 6775 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
6 | 4, 5 | sylibr 134 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 ∃wrex 2466 class class class wbr 4015 ωcom 4601 ≈ cen 6752 Fincfn 6754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-en 6755 df-fin 6757 |
This theorem is referenced by: dif1en 6893 0fin 6898 findcard2 6903 findcard2s 6904 diffisn 6907 pw1fin 6924 en1eqsn 6961 nninfwlpoimlemg 7187 nninfwlpoimlemginf 7188 exmidonfinlem 7206 fzfig 10444 hashennnuni 10773 hashennn 10774 hashun 10799 hashp1i 10804 unct 12457 xpsfrnel 12782 pwf1oexmid 15103 |
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