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| Mirrors > Home > MPE Home > Th. List > onfin | Structured version Visualization version GIF version | ||
| Description: An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92. (Contributed by NM, 26-Jul-2004.) |
| Ref | Expression |
|---|---|
| onfin | ⊢ (𝐴 ∈ On → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isfi 9016 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
| 2 | onomeneq 9265 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 ↔ 𝐴 = 𝑥)) | |
| 3 | eleq1a 2836 | . . . . . 6 ⊢ (𝑥 ∈ ω → (𝐴 = 𝑥 → 𝐴 ∈ ω)) | |
| 4 | 3 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 = 𝑥 → 𝐴 ∈ ω)) |
| 5 | 2, 4 | sylbid 240 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → 𝐴 ∈ ω)) |
| 6 | 5 | rexlimdva 3155 | . . 3 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ ω)) |
| 7 | enrefnn 9087 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) | |
| 8 | breq2 5147 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴)) | |
| 9 | 8 | rspcev 3622 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≈ 𝐴) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
| 10 | 7, 9 | mpdan 687 | . . 3 ⊢ (𝐴 ∈ ω → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
| 11 | 6, 10 | impbid1 225 | . 2 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ↔ 𝐴 ∈ ω)) |
| 12 | 1, 11 | bitrid 283 | 1 ⊢ (𝐴 ∈ On → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 Oncon0 6384 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-11 2157 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-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-mpt 5226 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-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 |
| This theorem is referenced by: onfin2 9268 fin17 10434 isfin7-2 10436 cantnfub 43334 tfsnfin 43365 |
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