Theorem onnminsb 2271

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 group(s):   x,A   ps,x

Proof of Theorem onnminsb

Step	Hyp	Ref	Expression
1		onnminsb.1	. . . . 5 |- (x = A -> (ph <-> ps))
2	1	elrab 1422	. . . 4 |- (A e. {x e. On | ph} <-> (A e. On /\ ps))
3		ssrab 1556	. . . . 5 |- {x e. On | ph} (_ On
4		onnmin 2270	. . . . 5 |- (({x e. On | ph} (_ On /\ A e. {x e. On | ph}) -> -. A e. |^|{x e. On | ph})
5	3, 4	mpan 518	. . . 4 |- (A e. {x e. On | ph} -> -. A e. |^|{x e. On | ph})
6	2, 5	sylbir 176	. . 3 |- ((A e. On /\ ps) -> -. A e. |^|{x e. On | ph})
7	6	exp 291	. 2 |- (A e. On -> (ps -> -. A e. |^|{x e. On | ph}))
8	7	con2d 83	1 |- (A e. On -> (A e. |^|{x e. On | ph} -> -. ps))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  {crab 1204   (_ wss 1487  |^|cint 1965  Oncon0 2199

This theorem is referenced by:  oawordeulem 3156  alephnbtwn 3674  cardaleph 3690

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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