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Theorem clos1conn 5879
Description: If a class is connected to an element of a closure via R, then it is a member of the closure. Theorem IX.5.14 of [Rosser] p. 246. (Contributed by SF, 13-Feb-2015.)
Hypothesis
Ref Expression
clos1base.1 C = Clos1 (S, R)
Assertion
Ref Expression
clos1conn ((A C ARB) → B C)

Proof of Theorem clos1conn
Dummy variables a x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brex 4689 . . 3 (ARB → (A V B V))
21adantl 452 . 2 ((A C ARB) → (A V B V))
3 eleq1 2413 . . . . 5 (x = A → (x CA C))
4 breq1 4642 . . . . 5 (x = A → (xRyARy))
53, 4anbi12d 691 . . . 4 (x = A → ((x C xRy) ↔ (A C ARy)))
65imbi1d 308 . . 3 (x = A → (((x C xRy) → y C) ↔ ((A C ARy) → y C)))
7 breq2 4643 . . . . 5 (y = B → (ARyARB))
87anbi2d 684 . . . 4 (y = B → ((A C ARy) ↔ (A C ARB)))
9 eleq1 2413 . . . 4 (y = B → (y CB C))
108, 9imbi12d 311 . . 3 (y = B → (((A C ARy) → y C) ↔ ((A C ARB) → B C)))
11 breq1 4642 . . . . . . . . . . . . . 14 (z = x → (zRyxRy))
1211rspcev 2955 . . . . . . . . . . . . 13 ((x a xRy) → z a zRy)
13 elima 4754 . . . . . . . . . . . . 13 (y (Ra) ↔ z a zRy)
1412, 13sylibr 203 . . . . . . . . . . . 12 ((x a xRy) → y (Ra))
1514ancoms 439 . . . . . . . . . . 11 ((xRy x a) → y (Ra))
16 ssel 3267 . . . . . . . . . . 11 ((Ra) a → (y (Ra) → y a))
1715, 16syl5 28 . . . . . . . . . 10 ((Ra) a → ((xRy x a) → y a))
1817exp3a 425 . . . . . . . . 9 ((Ra) a → (xRy → (x ay a)))
1918com12 27 . . . . . . . 8 (xRy → ((Ra) a → (x ay a)))
2019adantld 453 . . . . . . 7 (xRy → ((S a (Ra) a) → (x ay a)))
2120a2d 23 . . . . . 6 (xRy → (((S a (Ra) a) → x a) → ((S a (Ra) a) → y a)))
2221alimdv 1621 . . . . 5 (xRy → (a((S a (Ra) a) → x a) → a((S a (Ra) a) → y a)))
23 clos1base.1 . . . . . . . 8 C = Clos1 (S, R)
24 df-clos1 5873 . . . . . . . 8 Clos1 (S, R) = {a (S a (Ra) a)}
2523, 24eqtri 2373 . . . . . . 7 C = {a (S a (Ra) a)}
2625eleq2i 2417 . . . . . 6 (x Cx {a (S a (Ra) a)})
27 vex 2862 . . . . . . 7 x V
2827elintab 3937 . . . . . 6 (x {a (S a (Ra) a)} ↔ a((S a (Ra) a) → x a))
2926, 28bitri 240 . . . . 5 (x Ca((S a (Ra) a) → x a))
3025eleq2i 2417 . . . . . 6 (y Cy {a (S a (Ra) a)})
31 vex 2862 . . . . . . 7 y V
3231elintab 3937 . . . . . 6 (y {a (S a (Ra) a)} ↔ a((S a (Ra) a) → y a))
3330, 32bitri 240 . . . . 5 (y Ca((S a (Ra) a) → y a))
3422, 29, 333imtr4g 261 . . . 4 (xRy → (x Cy C))
3534impcom 419 . . 3 ((x C xRy) → y C)
366, 10, 35vtocl2g 2918 . 2 ((A V B V) → ((A C ARB) → B C))
372, 36mpcom 32 1 ((A C ARB) → B C)
Colors of variables: wff setvar class
Syntax hints:  wi 4   wa 358  wal 1540   = wceq 1642   wcel 1710  {cab 2339  wrex 2615  Vcvv 2859   wss 3257  cint 3926   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-br 4640  df-ima 4727  df-clos1 5873
This theorem is referenced by:  clos1induct  5880  clos1basesuc  5882  spaccl  6286  dmfrec  6316
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