Theorem limon 3090

Description: The class of ordinal numbers is a limit ordinal.

Assertion

Ref	Expression
limon	|- Lim On

Proof of Theorem limon

Step	Hyp	Ref	Expression
1		ordon 2983	. . 3 |- Ord On
2		onne0 3029	. . 3 |- On =/= (/)
3		unon 3084	. . . 4 |- U.On = On
4	3	eqcomi 1477	. . 3 |- On = U.On
5	1, 2, 4	3pm3.2i 817	. 2 |- (Ord On /\ On =/= (/) /\ On = U.On)
6		df-lim 2949	. 2 |- (Lim On <-> (Ord On /\ On =/= (/) /\ On = U.On))
7	5, 6	mpbir 190	1 |- Lim On

Syntax hints:   /\ w3a 774   = wceq 955   =/= wne 1583  (/)c0 2277  U.cuni 2499  Ord word 2943  Oncon0 2944  Lim wlim 2945

This theorem is referenced by:  limom 3142  limensuc 4496

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950

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