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Mirrors > Home > MPE Home > Th. List > ssnlim | Structured version Visualization version GIF version |
Description: An ordinal subclass of non-limit ordinals is a class of natural numbers. Exercise 7 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.) |
Ref | Expression |
---|---|
ssnlim | ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limom 7722 | . . . 4 ⊢ Lim ω | |
2 | ssel 3919 | . . . . 5 ⊢ (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → (ω ∈ 𝐴 → ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥})) | |
3 | limeq 6277 | . . . . . . . 8 ⊢ (𝑥 = ω → (Lim 𝑥 ↔ Lim ω)) | |
4 | 3 | notbid 318 | . . . . . . 7 ⊢ (𝑥 = ω → (¬ Lim 𝑥 ↔ ¬ Lim ω)) |
5 | 4 | elrab 3626 | . . . . . 6 ⊢ (ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ∈ On ∧ ¬ Lim ω)) |
6 | 5 | simprbi 497 | . . . . 5 ⊢ (ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → ¬ Lim ω) |
7 | 2, 6 | syl6 35 | . . . 4 ⊢ (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → (ω ∈ 𝐴 → ¬ Lim ω)) |
8 | 1, 7 | mt2i 137 | . . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → ¬ ω ∈ 𝐴) |
9 | 8 | adantl 482 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → ¬ ω ∈ 𝐴) |
10 | ordom 7716 | . . . 4 ⊢ Ord ω | |
11 | ordtri1 6298 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord ω) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴)) | |
12 | 10, 11 | mpan2 688 | . . 3 ⊢ (Ord 𝐴 → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴)) |
13 | 12 | adantr 481 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴)) |
14 | 9, 13 | mpbird 256 | 1 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {crab 3070 ⊆ wss 3892 Ord word 6264 Oncon0 6265 Lim wlim 6266 ωcom 7706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-tr 5197 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-om 7707 |
This theorem is referenced by: (None) |
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