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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 7876 | . . . . 5 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On) | |
2 | onfin 9255 | . . . . . 6 ⊢ (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω)) | |
3 | 2 | biimprcd 249 | . . . . 5 ⊢ (𝑥 ∈ ω → (𝑥 ∈ On → 𝑥 ∈ Fin)) |
4 | 1, 3 | jcai 516 | . . . 4 ⊢ (𝑥 ∈ ω → (𝑥 ∈ On ∧ 𝑥 ∈ Fin)) |
5 | 2 | biimpa 476 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ Fin) → 𝑥 ∈ ω) |
6 | 4, 5 | impbii 208 | . . 3 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Fin)) |
7 | elin 3963 | . . 3 ⊢ (𝑥 ∈ (On ∩ Fin) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Fin)) | |
8 | 6, 7 | bitr4i 278 | . 2 ⊢ (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ Fin)) |
9 | 8 | eqriv 2725 | 1 ⊢ ω = (On ∩ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 Oncon0 6369 ωcom 7870 Fincfn 8964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-om 7871 df-1o 8487 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 |
This theorem is referenced by: nnfiOLD 9257 cantnfcl 9691 ackbij1lem9 10252 ackbij1lem10 10253 ackbij1b 10263 sdom2en01 10326 fin23lem26 10349 fin56 10417 fin1a2lem9 10432 fzfi 13970 fz1isolem 14455 ackbijnn 15807 hauspwdom 23418 0finon 42878 1finon 42879 2finon 42880 3finon 42881 4finon 42882 finona1cl 42883 finonex 42884 dfom6 42961 |
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