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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 7588 | . . . . 5 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On) | |
2 | onfin 8711 | . . . . . 6 ⊢ (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω)) | |
3 | 2 | biimprcd 252 | . . . . 5 ⊢ (𝑥 ∈ ω → (𝑥 ∈ On → 𝑥 ∈ Fin)) |
4 | 1, 3 | jcai 519 | . . . 4 ⊢ (𝑥 ∈ ω → (𝑥 ∈ On ∧ 𝑥 ∈ Fin)) |
5 | 2 | biimpa 479 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ Fin) → 𝑥 ∈ ω) |
6 | 4, 5 | impbii 211 | . . 3 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Fin)) |
7 | elin 4171 | . . 3 ⊢ (𝑥 ∈ (On ∩ Fin) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Fin)) | |
8 | 6, 7 | bitr4i 280 | . 2 ⊢ (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ Fin)) |
9 | 8 | eqriv 2820 | 1 ⊢ ω = (On ∩ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 Oncon0 6193 ωcom 7582 Fincfn 8511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 |
This theorem is referenced by: nnfi 8713 cantnfcl 9132 ackbij1lem9 9652 ackbij1lem10 9653 ackbij1b 9663 sdom2en01 9726 fin23lem26 9749 fin56 9817 fin1a2lem9 9832 fzfi 13343 fz1isolem 13822 ackbijnn 15185 hauspwdom 22111 dfom6 39905 |
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