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Mirrors > Home > MPE Home > Th. List > limon | Structured version Visualization version GIF version |
Description: The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
Ref | Expression |
---|---|
limon | ⊢ Lim On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordon 7244 | . 2 ⊢ Ord On | |
2 | onn0 6027 | . 2 ⊢ On ≠ ∅ | |
3 | unon 7292 | . . 3 ⊢ ∪ On = On | |
4 | 3 | eqcomi 2834 | . 2 ⊢ On = ∪ On |
5 | df-lim 5968 | . 2 ⊢ (Lim On ↔ (Ord On ∧ On ≠ ∅ ∧ On = ∪ On)) | |
6 | 1, 2, 4, 5 | mpbir3an 1445 | 1 ⊢ Lim On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ≠ wne 2999 ∅c0 4144 ∪ cuni 4658 Ord word 5962 Oncon0 5963 Lim wlim 5964 |
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 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 |
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 4145 df-if 4307 df-pw 4380 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 |
This theorem is referenced by: limom 7341 oesuc 7874 limensuc 8406 limsucncmp 32967 |
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