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Theorem nchoicelem10 6298
 Description: Lemma for nchoice 6308. Stratification helper lemma. (Contributed by SF, 18-Mar-2015.)
Hypotheses
Ref Expression
nchoicelem10.1
nchoicelem10.2
Assertion
Ref Expression
nchoicelem10 Ins3 S Ins2 S S S Image1c Clos1

Proof of Theorem nchoicelem10
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nchoicelem10.2 . 2
2 elrn 4896 . . . . 5 S S S Image S S S Image
3 trtxp 5781 . . . . . . 7 S S S Image S S S Image
4 brcnv 4892 . . . . . . . . . 10 S S
5 df-br 4640 . . . . . . . . . 10 S S
6 snex 4111 . . . . . . . . . . . . 13
7 vex 2862 . . . . . . . . . . . . 13
86, 7opex 4588 . . . . . . . . . . . 12
98elcompl 3225 . . . . . . . . . . 11 S S
10 vex 2862 . . . . . . . . . . . 12
1110, 7opelssetsn 4760 . . . . . . . . . . 11 S
129, 11xchbinx 301 . . . . . . . . . 10 S
134, 5, 123bitri 262 . . . . . . . . 9 S
14 brres 4949 . . . . . . . . . 10 S S Image S S Image
15 brcnv 4892 . . . . . . . . . . . 12 S S
161, 7brsset 4758 . . . . . . . . . . . 12 S
1715, 16bitri 240 . . . . . . . . . . 11 S
18 elfix 5787 . . . . . . . . . . . 12 S Image S Image
19 brco 4883 . . . . . . . . . . . . 13 S Image Image S
20 vex 2862 . . . . . . . . . . . . . . . 16
217, 20brimage 5793 . . . . . . . . . . . . . . 15 Image
2220, 7brsset 4758 . . . . . . . . . . . . . . 15 S
2321, 22anbi12i 678 . . . . . . . . . . . . . 14 Image S
2423exbii 1582 . . . . . . . . . . . . 13 Image S
25 nchoicelem10.1 . . . . . . . . . . . . . . 15
2625, 7imaex 4747 . . . . . . . . . . . . . 14
27 sseq1 3292 . . . . . . . . . . . . . 14
2826, 27ceqsexv 2894 . . . . . . . . . . . . 13
2919, 24, 283bitri 262 . . . . . . . . . . . 12 S Image
3018, 29bitri 240 . . . . . . . . . . 11 S Image
3117, 30anbi12i 678 . . . . . . . . . 10 S S Image
3214, 31bitri 240 . . . . . . . . 9 S S Image
3313, 32anbi12i 678 . . . . . . . 8 S S S Image
34 ancom 437 . . . . . . . 8
3533, 34bitri 240 . . . . . . 7 S S S Image
36 annim 414 . . . . . . 7
373, 35, 363bitri 262 . . . . . 6 S S S Image
3837exbii 1582 . . . . 5 S S S Image
39 exnal 1574 . . . . 5
402, 38, 393bitrri 263 . . . 4 S S S Image
4140con1bii 321 . . 3 S S S Image
426, 1opex 4588 . . . 4
4342elcompl 3225 . . 3 S S S Image S S S Image
44 df-clos1 5873 . . . . 5 Clos1
4544eleq2i 2417 . . . 4 Clos1
4610elintab 3937 . . . 4
4745, 46bitri 240 . . 3 Clos1
4841, 43, 473bitr4i 268 . 2 S S S Image Clos1
491, 48releqel 5807 1 Ins3 S Ins2 S S S Image1c Clos1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358  wal 1540  wex 1541   wceq 1642   wcel 1710  cab 2339  cvv 2859   ∼ ccompl 3205   csymdif 3209   wss 3257  csn 3737  cint 3926  1cc1c 4134  cop 4561   class class class wbr 4639   S csset 4719   ccom 4721  cima 4722  ccnv 4771   crn 4773   cres 4774   ctxp 5735  cfix 5739   Ins2 cins2 5749   Ins3 cins3 5751  Imagecimage 5753   Clos1 cclos1 5872 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-1st 4723  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-res 4788  df-2nd 4797  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-clos1 5873 This theorem is referenced by:  nchoicelem11  6299  nchoicelem16  6304  nchoicelem18  6306
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