Theorem onminex 3015

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 groups:   x,y   ph,y   ps,x

Proof of Theorem onminex

Step	Hyp	Ref	Expression
1		hbab1 1464	. . . . . . 7 |- (y e. {x | (x e. On /\ ph)} -> A.x y e. {x | (x e. On /\ ph)})
2	1	hbint 2538	. . . . . 6 |- (y e. |^|{x | (x e. On /\ ph)} -> A.x y e. |^|{x | (x e. On /\ ph)})
3	2, 1	hbel 1563	. . . . . . 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-17 969	. . . . . . . . 9 |- (-. ps -> A.x -. ps)
5	2, 4	hbim 1005	. . . . . . . 8 |- ((y e. |^|{x | (x e. On /\ ph)} -> -. ps) -> A.x(y e. |^|{x | (x e. On /\ ph)} -> -. ps))
6	5	hbal 1003	. . . . . . 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 1007	. . . . . 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 1531	. . . . . . 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 1532	. . . . . . . . 9 |- (x = |^|{x | (x e. On /\ ph)} -> (y e. x <-> y e. |^|{x | (x e. On /\ ph)}))
10	9	imbi1d 612	. . . . . . . 8 |- (x = |^|{x | (x e. On /\ ph)} -> ((y e. x -> -. ps) <-> (y e. |^|{x | (x e. On /\ ph)} -> -. ps)))
11	10	albidv 1276	. . . . . . 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 627	. . . . . 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 1857	. . . . 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 495	. . . 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		ssab2 2126	. . . . 5 |- {x | (x e. On /\ ph)} (_ On
16		onint 3001	. . . . 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 694	. . . 4 |- ({x | (x e. On /\ ph)} =/= (/) -> |^|{x | (x e. On /\ ph)} e. {x | (x e. On /\ ph)})
18		oninton 3007	. . . . . . . 8 |- (({x | (x e. On /\ ph)} (_ On /\ {x | (x e. On /\ ph)} =/= (/)) -> |^|{x | (x e. On /\ ph)} e. On)
19	15, 18	mpan 694	. . . . . . 7 |- ({x | (x e. On /\ ph)} =/= (/) -> |^|{x | (x e. On /\ ph)} e. On)
20		onelon 2967	. . . . . . . 8 |- ((|^|{x | (x e. On /\ ph)} e. On /\ y e. |^|{x | (x e. On /\ ph)}) -> y e. On)
21	20	ex 373	. . . . . . 7 |- (|^|{x | (x e. On /\ ph)} e. On -> (y e. |^|{x | (x e. On /\ ph)} -> y e. On))
22	19, 21	syl 10	. . . . . 6 |- ({x | (x e. On /\ ph)} =/= (/) -> (y e. |^|{x | (x e. On /\ ph)} -> y e. On))
23		visset 1809	. . . . . . . . . 10 |- y e. V
24		eleq1 1531	. . . . . . . . . . 11 |- (x = y -> (x e. On <-> y e. On))
25		onminex.1	. . . . . . . . . . 11 |- (x = y -> (ph <-> ps))
26	24, 25	anbi12d 627	. . . . . . . . . 10 |- (x = y -> ((x e. On /\ ph) <-> (y e. On /\ ps)))
27	23, 26	elab 1893	. . . . . . . . 9 |- (y e. {x | (x e. On /\ ph)} <-> (y e. On /\ ps))
28		onnmin 3010	. . . . . . . . . 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 694	. . . . . . . . 9 |- (y e. {x | (x e. On /\ ph)} -> -. y e. |^|{x | (x e. On /\ ph)})
30	27, 29	sylbir 201	. . . . . . . 8 |- ((y e. On /\ ps) -> -. y e. |^|{x | (x e. On /\ ph)})
31	30	ex 373	. . . . . . 7 |- (y e. On -> (ps -> -. y e. |^|{x | (x e. On /\ ph)}))
32	31	con2d 91	. . . . . 6 |- (y e. On -> (y e. |^|{x | (x e. On /\ ph)} -> -. ps))
33	22, 32	syli 54	. . . . 5 |- ({x | (x e. On /\ ph)} =/= (/) -> (y e. |^|{x | (x e. On /\ ph)} -> -. ps))
34	33	19.21aiv 1284	. . . 4 |- ({x | (x e. On /\ ph)} =/= (/) -> A.y(y e. |^|{x | (x e. On /\ ph)} -> -. ps))
35	14, 17, 34	sylanc 471	. . 3 |- ({x | (x e. On /\ ph)} =/= (/) -> E.x(x e. {x | (x e. On /\ ph)} /\ A.y(y e. x -> -. ps)))
36		abn0 2286	. . 3 |- ({x | (x e. On /\ ph)} =/= (/) <-> E.x(x e. On /\ ph))
37		abid 1463	. . . . . . 7 |- (x e. {x | (x e. On /\ ph)} <-> (x e. On /\ ph))
38	37	bicomi 172	. . . . . 6 |- ((x e. On /\ ph) <-> x e. {x | (x e. On /\ ph)})
39		df-ral 1646	. . . . . 6 |- (A.y e. x -. ps <-> A.y(y e. x -> -. ps))
40	38, 39	anbi12i 482	. . . . 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 439	. . . . 5 |- (((x e. On /\ ph) /\ A.y e. x -. ps) <-> (x e. On /\ (ph /\ A.y e. x -. ps)))
42	40, 41	bitr3 175	. . . 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	exbii 1049	. . 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 218	. 2 |- (E.x(x e. On /\ ph) -> E.x(x e. On /\ (ph /\ A.y e. x -. ps)))
45		df-rex 1647	. 2 |- (E.x e. On ph <-> E.x(x e. On /\ ph))
46		df-rex 1647	. 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 219	1 |- (E.x e. On ph -> E.x e. On (ph /\ A.y e. x -. ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  {cab 1461   =/= wne 1582  A.wral 1642  E.wrex 1643   (_ wss 2043  (/)c0 2276  |^|cint 2528  Oncon0 2943

This theorem is referenced by:  tz7.49 3950  zorn2lem7 4774

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845