Theorem onnminsb 3022

Description: An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. ps is the wff resulting from the substitution of A for x in wff ph.

Hypothesis

Ref	Expression
onnminsb.1	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
onnminsb	|- (A e. On -> (A e. |^|{x e. On | ph} -> -. ps))

Distinct variable groups:   x,A   ps,x

Proof of Theorem onnminsb

Step	Hyp	Ref	Expression
1		onnminsb.1	. . . . 5 |- (x = A -> (ph <-> ps))
2	1	elrab 1908	. . . 4 |- (A e. {x e. On | ph} <-> (A e. On /\ ps))
3		ssrab2 2134	. . . . 5 |- {x e. On | ph} (_ On
4		onnmin 3021	. . . . 5 |- (({x e. On | ph} (_ On /\ A e. {x e. On | ph}) -> -. A e. |^|{x e. On | ph})
5	3, 4	mpan 697	. . . 4 |- (A e. {x e. On | ph} -> -. A e. |^|{x e. On | ph})
6	2, 5	sylbir 201	. . 3 |- ((A e. On /\ ps) -> -. A e. |^|{x e. On | ph})
7	6	ex 373	. 2 |- (A e. On -> (ps -> -. A e. |^|{x e. On | ph}))
8	7	con2d 91	1 |- (A e. On -> (A e. |^|{x e. On | ph} -> -. ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  {crab 1651   (_ wss 2050  |^|cint 2537  Oncon0 2954

This theorem is referenced by:  oawordeulem 4194  alephnbtwn 4879  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

Copyright terms: Public domain