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 6961 | . 2 ⊢ ω ⊆ On | |
2 | nnlim 6970 | . . 3 ⊢ (𝑥 ∈ ω → ¬ Lim 𝑥) | |
3 | 2 | rgen 2906 | . 2 ⊢ ∀𝑥 ∈ ω ¬ Lim 𝑥 |
4 | ssrab 3643 | . 2 ⊢ (ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω ¬ Lim 𝑥)) | |
5 | 1, 3, 4 | mpbir2an 957 | 1 ⊢ ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wral 2896 {crab 2900 ⊆ wss 3540 Oncon0 5640 Lim wlim 5641 ωcom 6957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |