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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 7759 | . 2 ⊢ Ord On | |
2 | onn0 6426 | . 2 ⊢ On ≠ ∅ | |
3 | unon 7814 | . . 3 ⊢ ∪ On = On | |
4 | 3 | eqcomi 2742 | . 2 ⊢ On = ∪ On |
5 | df-lim 6366 | . 2 ⊢ (Lim On ↔ (Ord On ∧ On ≠ ∅ ∧ On = ∪ On)) | |
6 | 1, 2, 4, 5 | mpbir3an 1342 | 1 ⊢ Lim On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ≠ wne 2941 ∅c0 4321 ∪ cuni 4907 Ord word 6360 Oncon0 6361 Lim wlim 6362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 |
This theorem is referenced by: limom 7866 oesuc 8522 limensuc 9150 limsucncmp 35269 dflim5 42012 |
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