Theorem onminsb 3015

Description: If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228.

Hypotheses

Ref	Expression
onminsb.1	|- (ps -> A.xps)
onminsb.2	|- (x = |^|{x e. On | ph} -> (ph <-> ps))

Assertion

Ref	Expression
onminsb	|- (E.x e. On ph -> ps)

Proof of Theorem onminsb

Step	Hyp	Ref	Expression
1		rabn0 2296	. . 3 |- ({x e. On | ph} =/= (/) <-> E.x e. On ph)
2		ssrab2 2134	. . . 4 |- {x e. On | ph} (_ On
3		onint 3012	. . . 4 |- (({x e. On | ph} (_ On /\ {x e. On | ph} =/= (/)) -> |^|{x e. On | ph} e. {x e. On | ph})
4	2, 3	mpan 697	. . 3 |- ({x e. On | ph} =/= (/) -> |^|{x e. On | ph} e. {x e. On | ph})
5	1, 4	sylbir 201	. 2 |- (E.x e. On ph -> |^|{x e. On | ph} e. {x e. On | ph})
6		hbrab1 1775	. . . . 5 |- (y e. {x e. On | ph} -> A.x y e. {x e. On | ph})
7	6	hbint 2547	. . . 4 |- (y e. |^|{x e. On | ph} -> A.x y e. |^|{x e. On | ph})
8		ax-17 973	. . . 4 |- (y e. On -> A.x y e. On)
9		onminsb.1	. . . 4 |- (ps -> A.xps)
10		onminsb.2	. . . 4 |- (x = |^|{x e. On | ph} -> (ph <-> ps))
11	7, 8, 9, 10	elrabf 1907	. . 3 |- (|^|{x e. On | ph} e. {x e. On | ph} <-> (|^|{x e. On | ph} e. On /\ ps))
12	11	pm3.27bi 326	. 2 |- (|^|{x e. On | ph} e. {x e. On | ph} -> ps)
13	5, 12	syl 10	1 |- (E.x e. On ph -> ps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 956   = wceq 958   e. wcel 960   =/= wne 1588  E.wrex 1649  {crab 1651   (_ wss 2050  (/)c0 2283  |^|cint 2537  Oncon0 2954

This theorem is referenced by:  oawordeulem 4194  rankid 4682  cardmin 4871  alephordlem1 4883  cardaleph 4896

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-int 2538  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

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