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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 7330 | . 2 ⊢ ω ⊆ On | |
2 | nnlim 7339 | . . 3 ⊢ (𝑥 ∈ ω → ¬ Lim 𝑥) | |
3 | 2 | rgen 3131 | . 2 ⊢ ∀𝑥 ∈ ω ¬ Lim 𝑥 |
4 | ssrab 3905 | . 2 ⊢ (ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω ¬ Lim 𝑥)) | |
5 | 1, 3, 4 | mpbir2an 704 | 1 ⊢ ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wral 3117 {crab 3121 ⊆ wss 3798 Oncon0 5963 Lim wlim 5964 ωcom 7326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 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 4145 df-if 4307 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-tr 4976 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-om 7327 |
This theorem is referenced by: (None) |
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