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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 8923 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | onomeneq 9179 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 ↔ 𝐴 = 𝑥)) | |
3 | eleq1a 2833 | . . . . . 6 ⊢ (𝑥 ∈ ω → (𝐴 = 𝑥 → 𝐴 ∈ ω)) | |
4 | 3 | adantl 483 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 = 𝑥 → 𝐴 ∈ ω)) |
5 | 2, 4 | sylbid 239 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → 𝐴 ∈ ω)) |
6 | 5 | rexlimdva 3153 | . . 3 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ ω)) |
7 | enrefnn 8998 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) | |
8 | breq2 5114 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴)) | |
9 | 8 | rspcev 3584 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≈ 𝐴) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
10 | 7, 9 | mpdan 686 | . . 3 ⊢ (𝐴 ∈ ω → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
11 | 6, 10 | impbid1 224 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3074 class class class wbr 5110 Oncon0 6322 ωcom 7807 ≈ cen 8887 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: onfin2 9182 fin17 10337 isfin7-2 10339 cantnfub 41685 |
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