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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 7798 | . 2 ⊢ Ord On | |
| 2 | onn0 6448 | . 2 ⊢ On ≠ ∅ | |
| 3 | unon 7852 | . . 3 ⊢ ∪ On = On | |
| 4 | 3 | eqcomi 2745 | . 2 ⊢ On = ∪ On | 
| 5 | df-lim 6388 | . 2 ⊢ (Lim On ↔ (Ord On ∧ On ≠ ∅ ∧ On = ∪ On)) | |
| 6 | 1, 2, 4, 5 | mpbir3an 1341 | 1 ⊢ Lim On | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ≠ wne 2939 ∅c0 4332 ∪ cuni 4906 Ord word 6382 Oncon0 6383 Lim wlim 6384 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 | 
| This theorem is referenced by: limom 7904 oesuc 8566 limensuc 9195 limsucncmp 36448 dflim5 43347 | 
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