![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > omon | Structured version Visualization version GIF version |
Description: The class of natural numbers ω is either an ordinal number (if we accept the Axiom of Infinity) or the proper class of all ordinal numbers (if we deny the Axiom of Infinity). Remark in [TakeutiZaring] p. 43. (Contributed by NM, 10-May-1998.) |
Ref | Expression |
---|---|
omon | ⊢ (ω ∈ On ∨ ω = On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 7308 | . 2 ⊢ Ord ω | |
2 | ordeleqon 7222 | . 2 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On)) | |
3 | 1, 2 | mpbi 222 | 1 ⊢ (ω ∈ On ∨ ω = On) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 874 = wceq 1653 ∈ wcel 2157 Ord word 5940 Oncon0 5941 ωcom 7299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-tr 4946 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-om 7300 |
This theorem is referenced by: omelon2 7311 infensuc 8380 elhf2 32795 |
Copyright terms: Public domain | W3C validator |