Theorem weds 5939

Description: Any property that holds for some element of a well-ordered set A has an R minimal element satisfying that property. (Contributed by SF, 20-Mar-2015.)

Hypotheses

Ref	Expression
weds.1	|- {x | ps} e. _V
weds.2	|- (x = y -> (ps <-> ch))
weds.3	|- (x = z -> (ps <-> th))
weds.4	|- (ph -> R We A)
weds.5	|- (ph -> E.x e. A ps)

Assertion

Ref	Expression
weds	|- (ph -> E.y e. A (ch /\ A.z e. A (th -> yRz)))

Distinct variable groups:   x,A,y,z   ch,x   ph,y,z   ps,y,z   y,R,z   th,x   x,y,z

Allowed substitution hints:   ph(x)   ps(x)   ch(y,z)   th(y,z)   R(x)

Proof of Theorem weds

Step	Hyp	Ref	Expression
1		weds.1	. . 3 |- {x | ps} e. _V
2		weds.2	. . 3 |- (x = y -> (ps <-> ch))
3		weds.3	. . 3 |- (x = z -> (ps <-> th))
4		weds.4	. . . 4 |- (ph -> R We A)
5		df-we 5907	. . . . . . 7 |- We = ( Or i^i Fr )
6	5	breqi 4646	. . . . . 6 |- (R We A <-> R( Or i^i Fr )A)
7		brin 4694	. . . . . 6 |- (R( Or i^i Fr )A <-> (R Or A /\ R Fr A))
8	6, 7	bitri 240	. . . . 5 |- (R We A <-> (R Or A /\ R Fr A))
9	8	simprbi 450	. . . 4 |- (R We A -> R Fr A)
10	4, 9	syl 15	. . 3 |- (ph -> R Fr A)
11		weds.5	. . 3 |- (ph -> E.x e. A ps)
12	1, 2, 3, 10, 11	frds 5936	. 2 |- (ph -> E.y e. A (ch /\ A.z e. A ((th /\ zRy) -> z = y)))
13		impexp 433	. . . . . . 7 |- (((th /\ zRy) -> z = y) <-> (th -> (zRy -> z = y)))
14	8	simplbi 446	. . . . . . . . . . . . 13 |- (R We A -> R Or A)
15	4, 14	syl 15	. . . . . . . . . . . 12 |- (ph -> R Or A)
16		sopc 5935	. . . . . . . . . . . . 13 |- (R Or A <-> (R Po A /\ R Connex A))
17	16	simprbi 450	. . . . . . . . . . . 12 |- (R Or A -> R Connex A)
18	15, 17	syl 15	. . . . . . . . . . 11 |- (ph -> R Connex A)
19	18	adantr 451	. . . . . . . . . 10 |- ((ph /\ (y e. A /\ z e. A)) -> R Connex A)
20		simprl 732	. . . . . . . . . 10 |- ((ph /\ (y e. A /\ z e. A)) -> y e. A)
21		simprr 733	. . . . . . . . . 10 |- ((ph /\ (y e. A /\ z e. A)) -> z e. A)
22	19, 20, 21	connexd 5932	. . . . . . . . 9 |- ((ph /\ (y e. A /\ z e. A)) -> (yRz \/ zRy))
23		ax1 1431	. . . . . . . . . . 11 |- (yRz -> ((zRy -> z = y) -> yRz))
24	23	a1i 10	. . . . . . . . . 10 |- ((ph /\ (y e. A /\ z e. A)) -> (yRz -> ((zRy -> z = y) -> yRz)))
25		pm2.27 35	. . . . . . . . . . 11 |- (zRy -> ((zRy -> z = y) -> z = y))
26		porta 5934	. . . . . . . . . . . . . . . . . . 19 |- (R Po A <-> (R Ref A /\ R Trans A /\ R Antisym A))
27	26	simp1bi 970	. . . . . . . . . . . . . . . . . 18 |- (R Po A -> R Ref A)
28	27	adantr 451	. . . . . . . . . . . . . . . . 17 |- ((R Po A /\ R Connex A) -> R Ref A)
29	16, 28	sylbi 187	. . . . . . . . . . . . . . . 16 |- (R Or A -> R Ref A)
30	15, 29	syl 15	. . . . . . . . . . . . . . 15 |- (ph -> R Ref A)
31	30	adantr 451	. . . . . . . . . . . . . 14 |- ((ph /\ z e. A) -> R Ref A)
32		simpr 447	. . . . . . . . . . . . . 14 |- ((ph /\ z e. A) -> z e. A)
33	31, 32	refd 5928	. . . . . . . . . . . . 13 |- ((ph /\ z e. A) -> zRz)
34	33	adantrl 696	. . . . . . . . . . . 12 |- ((ph /\ (y e. A /\ z e. A)) -> zRz)
35		breq1 4643	. . . . . . . . . . . 12 |- (z = y -> (zRz <-> yRz))
36	34, 35	syl5ibcom 211	. . . . . . . . . . 11 |- ((ph /\ (y e. A /\ z e. A)) -> (z = y -> yRz))
37	25, 36	syl9r 67	. . . . . . . . . 10 |- ((ph /\ (y e. A /\ z e. A)) -> (zRy -> ((zRy -> z = y) -> yRz)))
38	24, 37	jaod 369	. . . . . . . . 9 |- ((ph /\ (y e. A /\ z e. A)) -> ((yRz \/ zRy) -> ((zRy -> z = y) -> yRz)))
39	22, 38	mpd 14	. . . . . . . 8 |- ((ph /\ (y e. A /\ z e. A)) -> ((zRy -> z = y) -> yRz))
40	39	imim2d 48	. . . . . . 7 |- ((ph /\ (y e. A /\ z e. A)) -> ((th -> (zRy -> z = y)) -> (th -> yRz)))
41	13, 40	syl5bi 208	. . . . . 6 |- ((ph /\ (y e. A /\ z e. A)) -> (((th /\ zRy) -> z = y) -> (th -> yRz)))
42	41	anassrs 629	. . . . 5 |- (((ph /\ y e. A) /\ z e. A) -> (((th /\ zRy) -> z = y) -> (th -> yRz)))
43	42	ralimdva 2693	. . . 4 |- ((ph /\ y e. A) -> (A.z e. A ((th /\ zRy) -> z = y) -> A.z e. A (th -> yRz)))
44	43	anim2d 548	. . 3 |- ((ph /\ y e. A) -> ((ch /\ A.z e. A ((th /\ zRy) -> z = y)) -> (ch /\ A.z e. A (th -> yRz))))
45	44	reximdva 2727	. 2 |- (ph -> (E.y e. A (ch /\ A.z e. A ((th /\ zRy) -> z = y)) -> E.y e. A (ch /\ A.z e. A (th -> yRz))))
46	12, 45	mpd 14	1 |- (ph -> E.y e. A (ch /\ A.z e. A (th -> yRz)))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   e. wcel 1710  {cab 2339  A.wral 2615  E.wrex 2616  _Vcvv 2860   i^i cin 3209   class class class wbr 4640   Trans ctrans 5889   Ref cref 5890   Antisym cantisym 5891   Po cpartial 5892   Connex cconnex 5893   Or cstrict 5894   Fr cfound 5895   We cwe 5896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-meredith 1406  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ref 5901  df-partial 5903  df-connex 5904  df-strict 5905  df-found 5906  df-we 5907

This theorem is referenced by:  nchoicelem19  6308