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Mirrors > Home > MPE Home > Th. List > onfin2 | Structured version Visualization version GIF version |
Description: A set is a natural number iff it is a finite ordinal. (Contributed by Mario Carneiro, 22-Jan-2013.) |
Ref | Expression |
---|---|
onfin2 | ⊢ ω = (On ∩ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7813 | . . . . 5 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On) | |
2 | onfin 9181 | . . . . . 6 ⊢ (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω)) | |
3 | 2 | biimprcd 250 | . . . . 5 ⊢ (𝑥 ∈ ω → (𝑥 ∈ On → 𝑥 ∈ Fin)) |
4 | 1, 3 | jcai 518 | . . . 4 ⊢ (𝑥 ∈ ω → (𝑥 ∈ On ∧ 𝑥 ∈ Fin)) |
5 | 2 | biimpa 478 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ Fin) → 𝑥 ∈ ω) |
6 | 4, 5 | impbii 208 | . . 3 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Fin)) |
7 | elin 3931 | . . 3 ⊢ (𝑥 ∈ (On ∩ Fin) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Fin)) | |
8 | 6, 7 | bitr4i 278 | . 2 ⊢ (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ Fin)) |
9 | 8 | eqriv 2734 | 1 ⊢ ω = (On ∩ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3914 Oncon0 6322 ωcom 7807 Fincfn 8890 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-om 7808 df-1o 8417 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 |
This theorem is referenced by: nnfiOLD 9183 cantnfcl 9610 ackbij1lem9 10171 ackbij1lem10 10172 ackbij1b 10182 sdom2en01 10245 fin23lem26 10268 fin56 10336 fin1a2lem9 10351 fzfi 13884 fz1isolem 14367 ackbijnn 15720 hauspwdom 22868 0finon 41794 1finon 41795 2finon 41796 3finon 41797 4finon 41798 finona1cl 41799 finonex 41800 dfom6 41877 |
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