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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 R. Theorem IX.5.16 of [Rosser] p. 248. (Contributed by SF, 13-Feb-2015.)
Hypotheses
Ref Expression
clos1basesuc.1 S V
clos1basesuc.2 R V
clos1basesuc.3 C = Clos1 (S, R)
Assertion
Ref Expression
clos1basesuc (A C ↔ (A S x C xRA))
Distinct variable groups:   x,A   x,C   x,R
Allowed substitution hint:   S(x)

Proof of Theorem clos1basesuc
Dummy variables y w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clos1basesuc.1 . . 3 S V
2 clos1basesuc.2 . . 3 R V
3 clos1basesuc.3 . . 3 C = Clos1 (S, R)
4 abid2 2470 . . . . . . 7 {y y S} = S
54eqcomi 2357 . . . . . 6 S = {y y S}
6 df-ima 4727 . . . . . 6 (RC) = {y x C xRy}
75, 6uneq12i 3416 . . . . 5 (S ∪ (RC)) = ({y y S} ∪ {y x C xRy})
8 unab 3521 . . . . 5 ({y y S} ∪ {y x C xRy}) = {y (y S x C xRy)}
97, 8eqtri 2373 . . . 4 (S ∪ (RC)) = {y (y S x C xRy)}
101, 2clos1ex 5876 . . . . . . 7 Clos1 (S, R) V
113, 10eqeltri 2423 . . . . . 6 C V
122, 11imaex 4747 . . . . 5 (RC) V
131, 12unex 4106 . . . 4 (S ∪ (RC)) V
149, 13eqeltrri 2424 . . 3 {y (y S x C xRy)} V
15 eleq1 2413 . . . 4 (y = z → (y Sz S))
16 breq2 4643 . . . . 5 (y = z → (xRyxRz))
1716rexbidv 2635 . . . 4 (y = z → (x C xRyx C xRz))
1815, 17orbi12d 690 . . 3 (y = z → ((y S x C xRy) ↔ (z S x C xRz)))
19 eleq1 2413 . . . 4 (y = w → (y Sw S))
20 breq2 4643 . . . . 5 (y = w → (xRyxRw))
2120rexbidv 2635 . . . 4 (y = w → (x C xRyx C xRw))
2219, 21orbi12d 690 . . 3 (y = w → ((y S x C xRy) ↔ (w S x C xRw)))
23 eleq1 2413 . . . 4 (y = A → (y SA S))
24 breq2 4643 . . . . 5 (y = A → (xRyxRA))
2524rexbidv 2635 . . . 4 (y = A → (x C xRyx C xRA))
2623, 25orbi12d 690 . . 3 (y = A → ((y S x C xRy) ↔ (A S x C xRA)))
27 orc 374 . . 3 (y S → (y S x C xRy))
283clos1base 5878 . . . . . . . . . 10 S C
2928sseli 3269 . . . . . . . . 9 (z Sz C)
30 breq1 4642 . . . . . . . . . . 11 (y = z → (yRwzRw))
3130rspcev 2955 . . . . . . . . . 10 ((z C zRw) → y C yRw)
3231ex 423 . . . . . . . . 9 (z C → (zRwy C yRw))
3329, 32syl 15 . . . . . . . 8 (z S → (zRwy C yRw))
343clos1conn 5879 . . . . . . . . . 10 ((x C xRz) → z C)
3534, 32syl 15 . . . . . . . . 9 ((x C xRz) → (zRwy C yRw))
3635rexlimiva 2733 . . . . . . . 8 (x C xRz → (zRwy C yRw))
3733, 36jaoi 368 . . . . . . 7 ((z S x C xRz) → (zRwy C yRw))
3837impcom 419 . . . . . 6 ((zRw (z S x C xRz)) → y C yRw)
39 breq1 4642 . . . . . . 7 (x = y → (xRwyRw))
4039cbvrexv 2836 . . . . . 6 (x C xRwy C yRw)
4138, 40sylibr 203 . . . . 5 ((zRw (z S x C xRz)) → x C xRw)
4241olcd 382 . . . 4 ((zRw (z S x C xRz)) → (w S x C xRw))
43423adant1 973 . . 3 ((z C zRw (z S x C xRz)) → (w S x C xRw))
441, 2, 3, 14, 18, 22, 26, 27, 43clos1is 5881 . 2 (A C → (A S x C xRA))
4528sseli 3269 . . 3 (A SA C)
463clos1conn 5879 . . . 4 ((x C xRA) → A C)
4746rexlimiva 2733 . . 3 (x C xRAA C)
4845, 47jaoi 368 . 2 ((A S x C xRA) → A C)
4944, 48impbii 180 1 (A C ↔ (A S x C xRA))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 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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