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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 7240 | . 2 ⊢ Ord ω | |
2 | ordeleqon 7154 | . 2 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On)) | |
3 | 1, 2 | mpbi 220 | 1 ⊢ (ω ∈ On ∨ ω = On) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 = wceq 1632 ∈ wcel 2139 Ord word 5883 Oncon0 5884 ωcom 7231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-tr 4905 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-om 7232 |
This theorem is referenced by: omelon2 7243 infensuc 8305 elhf2 32609 |
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