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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 7492 | . 2 ⊢ Ord On | |
2 | onn0 6249 | . 2 ⊢ On ≠ ∅ | |
3 | unon 7540 | . . 3 ⊢ ∪ On = On | |
4 | 3 | eqcomi 2830 | . 2 ⊢ On = ∪ On |
5 | df-lim 6190 | . 2 ⊢ (Lim On ↔ (Ord On ∧ On ≠ ∅ ∧ On = ∪ On)) | |
6 | 1, 2, 4, 5 | mpbir3an 1337 | 1 ⊢ Lim On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ≠ wne 3016 ∅c0 4290 ∪ cuni 4831 Ord word 6184 Oncon0 6185 Lim wlim 6186 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 |
This theorem is referenced by: limom 7589 oesuc 8146 limensuc 8688 limsucncmp 33789 |
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