Theorem ceexlem1 6173

Description: Lemma for ceex 6174. Set up part of the stratification. (Contributed by SF, 6-Mar-2015.)

Assertion

Ref	Expression
ceexlem1	|- (<.{{a}}, n>. e. ( S o. SI Pw1Fn ) <-> ~P1 a e. n)

Distinct variable group:   n,a

Proof of Theorem ceexlem1

Dummy variables t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4111	. . . . . . . 8 |- {a} e. _V
2	1	brsnsi1 5775	. . . . . . 7 |- ({{a}} SI Pw1Fn u <-> E.t(u = {t} /\ {a} Pw1Fn t))
3	2	anbi1i 676	. . . . . 6 |- (({{a}} SI Pw1Fn u /\ u S n) <-> (E.t(u = {t} /\ {a} Pw1Fn t) /\ u S n))
4		19.41v 1901	. . . . . 6 |- (E.t((u = {t} /\ {a} Pw1Fn t) /\ u S n) <-> (E.t(u = {t} /\ {a} Pw1Fn t) /\ u S n))
5		anass 630	. . . . . . 7 |- (((u = {t} /\ {a} Pw1Fn t) /\ u S n) <-> (u = {t} /\ ({a} Pw1Fn t /\ u S n)))
6	5	exbii 1582	. . . . . 6 |- (E.t((u = {t} /\ {a} Pw1Fn t) /\ u S n) <-> E.t(u = {t} /\ ({a} Pw1Fn t /\ u S n)))
7	3, 4, 6	3bitr2i 264	. . . . 5 |- (({{a}} SI Pw1Fn u /\ u S n) <-> E.t(u = {t} /\ ({a} Pw1Fn t /\ u S n)))
8	7	exbii 1582	. . . 4 |- (E.u({{a}} SI Pw1Fn u /\ u S n) <-> E.uE.t(u = {t} /\ ({a} Pw1Fn t /\ u S n)))
9		excom 1741	. . . 4 |- (E.uE.t(u = {t} /\ ({a} Pw1Fn t /\ u S n)) <-> E.tE.u(u = {t} /\ ({a} Pw1Fn t /\ u S n)))
10	8, 9	bitri 240	. . 3 |- (E.u({{a}} SI Pw1Fn u /\ u S n) <-> E.tE.u(u = {t} /\ ({a} Pw1Fn t /\ u S n)))
11		snex 4111	. . . . . 6 |- {t} e. _V
12		breq1 4642	. . . . . . 7 |- (u = {t} -> (u S n <-> {t} S n))
13	12	anbi2d 684	. . . . . 6 |- (u = {t} -> (({a} Pw1Fn t /\ u S n) <-> ({a} Pw1Fn t /\ {t} S n)))
14	11, 13	ceqsexv 2894	. . . . 5 |- (E.u(u = {t} /\ ({a} Pw1Fn t /\ u S n)) <-> ({a} Pw1Fn t /\ {t} S n))
15		vex 2862	. . . . . . 7 |- a e. _V
16	15	brpw1fn 5854	. . . . . 6 |- ({a} Pw1Fn t <-> t = ~P1 a)
17		vex 2862	. . . . . . 7 |- t e. _V
18		vex 2862	. . . . . . 7 |- n e. _V
19	17, 18	brssetsn 4759	. . . . . 6 |- ({t} S n <-> t e. n)
20	16, 19	anbi12i 678	. . . . 5 |- (({a} Pw1Fn t /\ {t} S n) <-> (t = ~P1 a /\ t e. n))
21	14, 20	bitri 240	. . . 4 |- (E.u(u = {t} /\ ({a} Pw1Fn t /\ u S n)) <-> (t = ~P1 a /\ t e. n))
22	21	exbii 1582	. . 3 |- (E.tE.u(u = {t} /\ ({a} Pw1Fn t /\ u S n)) <-> E.t(t = ~P1 a /\ t e. n))
23	10, 22	bitri 240	. 2 |- (E.u({{a}} SI Pw1Fn u /\ u S n) <-> E.t(t = ~P1 a /\ t e. n))
24		opelco 4884	. 2 |- (<.{{a}}, n>. e. ( S o. SI Pw1Fn ) <-> E.u({{a}} SI Pw1Fn u /\ u S n))
25		df-clel 2349	. 2 |- (~P1 a e. n <-> E.t(t = ~P1 a /\ t e. n))
26	23, 24, 25	3bitr4i 268	1 |- (<.{{a}}, n>. e. ( S o. SI Pw1Fn ) <-> ~P1 a e. n)

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {csn 3737  ~P₁ cpw1 4135  <.cop 4561   class class class wbr 4639   S csset 4719   SI csi 4720   o. ccom 4721   Pw1Fn cpw1fn 5765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-mpt 5652  df-pw1fn 5766

This theorem is referenced by:  ceex  6174