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Theorem clos1basesuc 5882
 Description: A member of a closure is either in the base set or connected to another member by . Theorem IX.5.16 of [Rosser] p. 248. (Contributed by SF, 13-Feb-2015.)
Hypotheses
Ref Expression
clos1basesuc.1
clos1basesuc.2
clos1basesuc.3 Clos1
Assertion
Ref Expression
clos1basesuc
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem clos1basesuc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clos1basesuc.1 . . 3
2 clos1basesuc.2 . . 3
3 clos1basesuc.3 . . 3 Clos1
4 abid2 2470 . . . . . . 7
54eqcomi 2357 . . . . . 6
6 df-ima 4727 . . . . . 6
75, 6uneq12i 3416 . . . . 5
8 unab 3521 . . . . 5
97, 8eqtri 2373 . . . 4
101, 2clos1ex 5876 . . . . . . 7 Clos1
113, 10eqeltri 2423 . . . . . 6
122, 11imaex 4747 . . . . 5
131, 12unex 4106 . . . 4
149, 13eqeltrri 2424 . . 3
15 eleq1 2413 . . . 4
16 breq2 4643 . . . . 5
1716rexbidv 2635 . . . 4
1815, 17orbi12d 690 . . 3
19 eleq1 2413 . . . 4
20 breq2 4643 . . . . 5
2120rexbidv 2635 . . . 4
2219, 21orbi12d 690 . . 3
23 eleq1 2413 . . . 4
24 breq2 4643 . . . . 5
2524rexbidv 2635 . . . 4
2623, 25orbi12d 690 . . 3
27 orc 374 . . 3
283clos1base 5878 . . . . . . . . . 10
2928sseli 3269 . . . . . . . . 9
30 breq1 4642 . . . . . . . . . . 11
3130rspcev 2955 . . . . . . . . . 10
3231ex 423 . . . . . . . . 9
3329, 32syl 15 . . . . . . . 8
343clos1conn 5879 . . . . . . . . . 10
3534, 32syl 15 . . . . . . . . 9
3635rexlimiva 2733 . . . . . . . 8
3733, 36jaoi 368 . . . . . . 7
3837impcom 419 . . . . . 6
39 breq1 4642 . . . . . . 7
4039cbvrexv 2836 . . . . . 6
4138, 40sylibr 203 . . . . 5
4241olcd 382 . . . 4
441, 2, 3, 14, 18, 22, 26, 27, 43clos1is 5881 . 2
4528sseli 3269 . . 3
463clos1conn 5879 . . . 4
4746rexlimiva 2733 . . 3
4845, 47jaoi 368 . 2
4944, 48impbii 180 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wo 357   wa 358   wceq 1642   wcel 1710  cab 2339  wrex 2615  cvv 2859   cun 3207   class class class wbr 4639  cima 4722   Clos1 cclos1 5872 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-swap 4724  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-dm 4787  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:  clos1baseima  5883  clos1basesucg  5884
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