Theorem onunisuc 3106

Description: An ordinal number is equal to the union of its successor.

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
onunisuc	|- U.suc A = A

Proof of Theorem onunisuc

Step	Hyp	Ref	Expression
1		on.1	. . 3 |- A e. On
2	1	ontrc 3096	. 2 |- Tr A
3	1	elisseti 1818	. . 3 |- A e. V
4	3	unisuc 3046	. 2 |- (Tr A <-> U.suc A = A)
5	2, 4	mpbi 189	1 |- U.suc A = A

Syntax hints:   = wceq 956   e. wcel 958  U.cuni 2503  Tr wtr 2680  Oncon0 2948  suc csuc 2950

This theorem is referenced by:  tz7.44-2 3929  rankuni 4698

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-tr 2681  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954

Copyright terms: Public domain