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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 7796 | . 2 ⊢ Ord On | |
2 | onn0 6451 | . 2 ⊢ On ≠ ∅ | |
3 | unon 7851 | . . 3 ⊢ ∪ On = On | |
4 | 3 | eqcomi 2744 | . 2 ⊢ On = ∪ On |
5 | df-lim 6391 | . 2 ⊢ (Lim On ↔ (Ord On ∧ On ≠ ∅ ∧ On = ∪ On)) | |
6 | 1, 2, 4, 5 | mpbir3an 1340 | 1 ⊢ Lim On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ≠ wne 2938 ∅c0 4339 ∪ cuni 4912 Ord word 6385 Oncon0 6386 Lim wlim 6387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 |
This theorem is referenced by: limom 7903 oesuc 8564 limensuc 9193 limsucncmp 36429 dflim5 43319 |
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