Theorem orduniss2 3090

Description: The union of the ordinal subsets of an ordinal number is that number.

Assertion

Ref	Expression
orduniss2	|- (Ord A -> U.{x e. On | x (_ A} = A)

Distinct variable group:   x,A

Proof of Theorem orduniss2

Step	Hyp	Ref	Expression
1		ordpwsuc 3066	. . . 4 |- (Ord A -> (P~A i^i On) = suc A)
2		df-rab 1652	. . . . 5 |- {x e. On | x (_ A} = {x | (x e. On /\ x (_ A)}
3		inab 2268	. . . . . 6 |- ({x | x e. On} i^i {x | x (_ A}) = {x | (x e. On /\ x (_ A)}
4		incom 2208	. . . . . 6 |- ({x | x e. On} i^i {x | x (_ A}) = ({x | x (_ A} i^i {x | x e. On})
5	3, 4	eqtr3 1497	. . . . 5 |- {x | (x e. On /\ x (_ A)} = ({x | x (_ A} i^i {x | x e. On})
6		df-pw 2402	. . . . . . 7 |- P~A = {x | x (_ A}
7	6	eqcomi 1479	. . . . . 6 |- {x | x (_ A} = P~A
8		abid2 1580	. . . . . 6 |- {x | x e. On} = On
9	7, 8	ineq12i 2215	. . . . 5 |- ({x | x (_ A} i^i {x | x e. On}) = (P~A i^i On)
10	2, 5, 9	3eqtr 1499	. . . 4 |- {x e. On | x (_ A} = (P~A i^i On)
11	1, 10	syl5eq 1519	. . 3 |- (Ord A -> {x e. On | x (_ A} = suc A)
12	11	unieqd 2512	. 2 |- (Ord A -> U.{x e. On | x (_ A} = U.suc A)
13		ordunisuc 3089	. 2 |- (Ord A -> U.suc A = A)
14	12, 13	eqtrd 1507	1 |- (Ord A -> U.{x e. On | x (_ A} = A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  {crab 1648   i^i cin 2046   (_ wss 2047  P~cpw 2401  U.cuni 2503  Ord word 2947  Oncon0 2948  suc csuc 2950

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-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  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