Theorem onminesb 3010

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

Assertion

Ref	Expression
onminesb	|- (E.x e. On ph -> [|^|{x e. On | ph} / x]ph)

Proof of Theorem onminesb

Step	Hyp	Ref	Expression
1		rabn0 2292	. . 3 |- ({x e. On | ph} =/= (/) <-> E.x e. On ph)
2		ssrab2 2131	. . . 4 |- {x e. On | ph} (_ On
3		onint 3006	. . . 4 |- (({x e. On | ph} (_ On /\ {x e. On | ph} =/= (/)) -> |^|{x e. On | ph} e. {x e. On | ph})
4	2, 3	mpan 695	. . 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		ax-17 971	. . . 4 |- (y e. On -> A.x y e. On)
7	6	elrabsf 1963	. . 3 |- (|^|{x e. On | ph} e. {x e. On | ph} <-> (|^|{x e. On | ph} e. On /\ [|^|{x e. On | ph} / x]ph))
8	7	pm3.27bi 326	. 2 |- (|^|{x e. On | ph} e. {x e. On | ph} -> [|^|{x e. On | ph} / x]ph)
9	5, 8	syl 10	1 |- (E.x e. On ph -> [|^|{x e. On | ph} / x]ph)

Syntax hints:   -> wi 3   e. wcel 958  [wsbc 1170   =/= wne 1585  E.wrex 1646  {crab 1648   (_ wss 2047  (/)c0 2280  |^|cint 2533  Oncon0 2948

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-9 965  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-sbc 1942  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-int 2534  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

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