Theorem orduniss2 3096

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 3072	. . . 4 |- (Ord A -> (P~A i^i On) = suc A)
2		df-rab 1655	. . . . 5 |- {x e. On | x (_ A} = {x | (x e. On /\ x (_ A)}
3		inab 2271	. . . . . 6 |- ({x | x e. On} i^i {x | x (_ A}) = {x | (x e. On /\ x (_ A)}
4		incom 2211	. . . . . 6 |- ({x | x e. On} i^i {x | x (_ A}) = ({x | x (_ A} i^i {x | x e. On})
5	3, 4	eqtr3 1500	. . . . 5 |- {x | (x e. On /\ x (_ A)} = ({x | x (_ A} i^i {x | x e. On})
6		df-pw 2406	. . . . . . 7 |- P~A = {x | x (_ A}
7	6	eqcomi 1482	. . . . . 6 |- {x | x (_ A} = P~A
8		abid2 1583	. . . . . 6 |- {x | x e. On} = On
9	7, 8	ineq12i 2218	. . . . 5 |- ({x | x (_ A} i^i {x | x e. On}) = (P~A i^i On)
10	2, 5, 9	3eqtr 1502	. . . 4 |- {x e. On | x (_ A} = (P~A i^i On)
11	1, 10	syl5eq 1522	. . 3 |- (Ord A -> {x e. On | x (_ A} = suc A)
12	11	unieqd 2516	. 2 |- (Ord A -> U.{x e. On | x (_ A} = U.suc A)
13		ordunisuc 3095	. 2 |- (Ord A -> U.suc A = A)
14	12, 13	eqtrd 1510	1 |- (Ord A -> U.{x e. On | x (_ A} = A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  {crab 1651   i^i cin 2049   (_ wss 2050  P~cpw 2405  U.cuni 2507  Ord word 2953  Oncon0 2954  suc csuc 2956

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-suc 2960

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