Theorem onminex 2275

Description: If a wff is true for an ordinal number, there is a smallest ordinal number for which it is true.

Hypothesis

Ref	Expression
onminex.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
onminex	|- (E.x e. On ph -> E.x e. On (ph /\ A.y e. x -. ps))

Distinct variable group(s):   x,y   ph,y   ps,x

Proof of Theorem onminex

Step	Hyp	Ref	Expression
1		hbab1 1095	. . . . . . 7 |- (y e. {x | (x e. On /\ ph)} -> A.x y e. {x | (x e. On /\ ph)})
2	1	hbint 1975	. . . . . 6 |- (y e. |^|{x | (x e. On /\ ph)} -> A.x y e. |^|{x | (x e. On /\ ph)})
3	2, 1	hbel 1172	. . . . . . 7 |- (|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} -> A.x|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)})
4		ax-17 925	. . . . . . . . 9 |- (-. ps -> A.x -. ps)
5	2, 4	hbim 702	. . . . . . . 8 |- ((y e. |^|{x | (x e. On /\ ph)} -> -. ps) -> A.x(y e. |^|{x | (x e. On /\ ph)} -> -. ps))
6	5	hbal 700	. . . . . . 7 |- (A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps) -> A.xA.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps))
7	3, 6	hban 704	. . . . . 6 |- ((|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} /\ A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps)) -> A.x(|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} /\ A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps)))
8		eleq1 1149	. . . . . . 7 |- (x = |^|{x | (x e. On /\ ph)} -> (x e. {x | (x e. On /\ ph)} <-> |^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)}))
9		eleq2 1150	. . . . . . . . 9 |- (x = |^|{x | (x e. On /\ ph)} -> (y e. x <-> y e. |^|{x | (x e. On /\ ph)}))
10	9	imbi1d 465	. . . . . . . 8 |- (x = |^|{x | (x e. On /\ ph)} -> ((y e. x -> -. ps) <-> (y e. |^|{x | (x e. On /\ ph)} -> -. ps)))
11	10	bialdv 935	. . . . . . 7 |- (x = |^|{x | (x e. On /\ ph)} -> (A.y(y e. x -> -. ps) <-> A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps)))
12	8, 11	anbi12d 476	. . . . . 6 |- (x = |^|{x | (x e. On /\ ph)} -> ((x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)) <-> (|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} /\ A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps))))
13	2, 7, 12	cla4egf 1395	. . . . 5 |- (|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} -> ((|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} /\ A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps)) -> E.x(x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps))))
14	13	anabsi5 377	. . . 4 |- ((|^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)} /\ A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps)) -> E.x(x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)))
15		ssab 1555	. . . . 5 |- {x | (x e. On /\ ph)} (_ On
16		onint 2261	. . . . 5 |- (({x | (x e. On /\ ph)} (_ On /\ -. {x | (x e. On /\ ph)} = (/)) -> |^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)})
17	15, 16	mpan 518	. . . 4 |- (-. {x | (x e. On /\ ph)} = (/) -> |^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)})
18		oninton 2267	. . . . . . . 8 |- (({x | (x e. On /\ ph)} (_ On /\ -. {x | (x e. On /\ ph)} = (/)) -> |^|{x | (x e. On /\ ph)} e. On)
19	15, 18	mpan 518	. . . . . . 7 |- (-. {x | (x e. On /\ ph)} = (/) -> |^|{x | (x e. On /\ ph)} e. On)
20		onelon 2223	. . . . . . . 8 |- ((|^|{x | (x e. On /\ ph)} e. On /\ y e. |^|{x | (x e. On /\ ph)}) -> y e. On)
21	20	exp 291	. . . . . . 7 |- (|^|{x | (x e. On /\ ph)} e. On -> (y e. |^|{x | (x e. On /\ ph)} -> y e. On))
22	19, 21	syl 12	. . . . . 6 |- (-. {x | (x e. On /\ ph)} = (/) -> (y e. |^|{x | (x e. On /\ ph)} -> y e. On))
23		visset 1350	. . . . . . . . . 10 |- y e. V
24		eleq1 1149	. . . . . . . . . . 11 |- (x = y -> (x e. On <-> y e. On))
25		onminex.1	. . . . . . . . . . 11 |- (x = y -> (ph <-> ps))
26	24, 25	anbi12d 476	. . . . . . . . . 10 |- (x = y -> ((x e. On /\ ph) <-> (y e. On /\ ps)))
27	23, 26	elab 1415	. . . . . . . . 9 |- (y e. {x | (x e. On /\ ph)} <-> (y e. On /\ ps))
28		onnmin 2270	. . . . . . . . . 10 |- (({x | (x e. On /\ ph)} (_ On /\ y e. {x | (x e. On /\ ph)}) -> -. y e. |^|{x | (x e. On /\ ph)})
29	15, 28	mpan 518	. . . . . . . . 9 |- (y e. {x | (x e. On /\ ph)} -> -. y e. |^|{x | (x e. On /\ ph)})
30	27, 29	sylbir 176	. . . . . . . 8 |- ((y e. On /\ ps) -> -. y e. |^|{x | (x e. On /\ ph)})
31	30	exp 291	. . . . . . 7 |- (y e. On -> (ps -> -. y e. |^|{x | (x e. On /\ ph)}))
32	31	con2d 83	. . . . . 6 |- (y e. On -> (y e. |^|{x | (x e. On /\ ph)} -> -. ps))
33	22, 32	syli 52	. . . . 5 |- (-. {x | (x e. On /\ ph)} = (/) -> (y e. |^|{x | (x e. On /\ ph)} -> -. ps))
34	33	19.21aiv 943	. . . 4 |- (-. {x | (x e. On /\ ph)} = (/) -> A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps))
35	14, 17, 34	sylanc 361	. . 3 |- (-. {x | (x e. On /\ ph)} = (/) -> E.x(x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)))
36		abn0 1715	. . 3 |- (-. {x | (x e. On /\ ph)} = (/) <-> E.x(x e. On /\ ph))
37		abid 1094	. . . . . . 7 |- (x e. {x | (x e. On /\ ph)} <-> (x e. On /\ ph))
38	37	bicomi 150	. . . . . 6 |- ((x e. On /\ ph) <-> x e. {x | (x e. On /\ ph)})
39		df-ral 1205	. . . . . 6 |- (A.y e. x -. ps <-> A.y(y e. x -> -. ps))
40	38, 39	anbi12i 369	. . . . 5 |- (((x e. On /\ ph) /\ A.y e. x -. ps) <-> (x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)))
41		anass 336	. . . . 5 |- (((x e. On /\ ph) /\ A.y e. x -. ps) <-> (x e. On /\ (ph /\ A.y e. x -. ps)))
42	40, 41	bitr3 153	. . . 4 |- ((x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)) <-> (x e. On /\ (ph /\ A.y e. x -. ps)))
43	42	biex 733	. . 3 |- (E.x(x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)) <-> E.x(x e. On /\ (ph /\ A.y e. x -. ps)))
44	35, 36, 43	3imtr3 191	. 2 |- (E.x(x e. On /\ ph) -> E.x(x e. On /\ (ph /\ A.y e. x -. ps)))
45		df-rex 1206	. 2 |- (E.x e. On ph <-> E.x(x e. On /\ ph))
46		df-rex 1206	. 2 |- (E.x e. On (ph /\ A.y e. x -. ps) <-> E.x(x e. On /\ (ph /\ A.y e. x -. ps)))
47	44, 45, 46	3imtr4 192	1 |- (E.x e. On ph -> E.x e. On (ph /\ A.y e. x -. ps))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   (_ wss 1487  (/)c0 1707  |^|cint 1965  Oncon0 2199

This theorem is referenced by:  tz7.49 2997  zornlem7 3609

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-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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