Theorem onminesb 3880

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 2893	. . 3 |- ({x e. On | ph} =/= (/) <-> E.x e. On ph)
2		ssrab2 2692	. . . 4 |- {x e. On | ph} C_ On
3		onint 3876	. . . 4 |- (({x e. On | ph} C_ On /\ {x e. On | ph} =/= (/)) -> |^|{x e. On | ph} e. {x e. On | ph})
4	2, 3	mpan 759	. . 3 |- ({x e. On | ph} =/= (/) -> |^|{x e. On | ph} e. {x e. On | ph})
5	1, 4	sylbir 218	. 2 |- (E.x e. On ph -> |^|{x e. On | ph} e. {x e. On | ph})
6		ax-17 1317	. . . 4 |- (y e. On -> A.x y e. On)
7	6	elrabsf 2486	. . 3 |- (|^|{x e. On | ph} e. {x e. On | ph} <-> (|^|{x e. On | ph} e. On /\ [|^|{x e. On | ph} / x]ph))
8	7	simprbi 353	. 2 |- (|^|{x e. On | ph} e. {x e. On | ph} -> [|^|{x e. On | ph} / x]ph)
9	5, 8	syl 12	1 |- (E.x e. On ph -> [|^|{x e. On | ph} / x]ph)

Syntax hints:   -> wi 3   e. wcel 1300  [wsbc 1534   =/= wne 2017  E.wrex 2106  {crab 2108   C_ wss 2593  (/)c0 2875  |^|cint 3214  Oncon0 3657

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

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