![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > omssnlim | Structured version Visualization version GIF version |
Description: The class of natural numbers is a subclass of the class of non-limit ordinal numbers. Exercise 4 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
omssnlim | ⊢ ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omsson 7335 | . 2 ⊢ ω ⊆ On | |
2 | nnlim 7344 | . . 3 ⊢ (𝑥 ∈ ω → ¬ Lim 𝑥) | |
3 | 2 | rgen 3131 | . 2 ⊢ ∀𝑥 ∈ ω ¬ Lim 𝑥 |
4 | ssrab 3907 | . 2 ⊢ (ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω ¬ Lim 𝑥)) | |
5 | 1, 3, 4 | mpbir2an 702 | 1 ⊢ ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wral 3117 {crab 3121 ⊆ wss 3798 Oncon0 5967 Lim wlim 5968 ωcom 7331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-tr 4978 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-om 7332 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |