Theorem clos1conn 5880

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.

Step	Hyp	Ref	Expression
1		brex 4690	. . 3 ⊢ (ARB → (A ∈ V ∧ B ∈ V))
2	1	adantl 452	. 2 ⊢ ((A ∈ C ∧ ARB) → (A ∈ V ∧ B ∈ V))
3		eleq1 2413	. . . . 5 ⊢ (x = A → (x ∈ C ↔ A ∈ C))
4		breq1 4643	. . . . 5 ⊢ (x = A → (xRy ↔ ARy))
5	3, 4	anbi12d 691	. . . 4 ⊢ (x = A → ((x ∈ C ∧ xRy) ↔ (A ∈ C ∧ ARy)))
6	5	imbi1d 308	. . 3 ⊢ (x = A → (((x ∈ C ∧ xRy) → y ∈ C) ↔ ((A ∈ C ∧ ARy) → y ∈ C)))
7		breq2 4644	. . . . 5 ⊢ (y = B → (ARy ↔ ARB))
8	7	anbi2d 684	. . . 4 ⊢ (y = B → ((A ∈ C ∧ ARy) ↔ (A ∈ C ∧ ARB)))
9		eleq1 2413	. . . 4 ⊢ (y = B → (y ∈ C ↔ B ∈ C))
10	8, 9	imbi12d 311	. . 3 ⊢ (y = B → (((A ∈ C ∧ ARy) → y ∈ C) ↔ ((A ∈ C ∧ ARB) → B ∈ C)))
11		breq1 4643	. . . . . . . . . . . . . 14 ⊢ (z = x → (zRy ↔ xRy))
12	11	rspcev 2956	. . . . . . . . . . . . 13 ⊢ ((x ∈ a ∧ xRy) → ∃z ∈ a zRy)
13		elima 4755	. . . . . . . . . . . . 13 ⊢ (y ∈ (R “ a) ↔ ∃z ∈ a zRy)
14	12, 13	sylibr 203	. . . . . . . . . . . 12 ⊢ ((x ∈ a ∧ xRy) → y ∈ (R “ a))
15	14	ancoms 439	. . . . . . . . . . 11 ⊢ ((xRy ∧ x ∈ a) → y ∈ (R “ a))
16		ssel 3268	. . . . . . . . . . 11 ⊢ ((R “ a) ⊆ a → (y ∈ (R “ a) → y ∈ a))
17	15, 16	syl5 28	. . . . . . . . . 10 ⊢ ((R “ a) ⊆ a → ((xRy ∧ x ∈ a) → y ∈ a))
18	17	exp3a 425	. . . . . . . . 9 ⊢ ((R “ a) ⊆ a → (xRy → (x ∈ a → y ∈ a)))
19	18	com12 27	. . . . . . . 8 ⊢ (xRy → ((R “ a) ⊆ a → (x ∈ a → y ∈ a)))
20	19	adantld 453	. . . . . . 7 ⊢ (xRy → ((S ⊆ a ∧ (R “ a) ⊆ a) → (x ∈ a → y ∈ a)))
21	20	a2d 23	. . . . . 6 ⊢ (xRy → (((S ⊆ a ∧ (R “ a) ⊆ a) → x ∈ a) → ((S ⊆ a ∧ (R “ a) ⊆ a) → y ∈ a)))
22	21	alimdv 1621	. . . . 5 ⊢ (xRy → (∀a((S ⊆ a ∧ (R “ a) ⊆ a) → x ∈ a) → ∀a((S ⊆ a ∧ (R “ a) ⊆ a) → y ∈ a)))
23		clos1base.1	. . . . . . . 8 ⊢ C = Clos1 (S, R)
24		df-clos1 5874	. . . . . . . 8 ⊢ Clos1 (S, R) = ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)}
25	23, 24	eqtri 2373	. . . . . . 7 ⊢ C = ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)}
26	25	eleq2i 2417	. . . . . 6 ⊢ (x ∈ C ↔ x ∈ ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)})
27		vex 2863	. . . . . . 7 ⊢ x ∈ V
28	27	elintab 3938	. . . . . 6 ⊢ (x ∈ ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)} ↔ ∀a((S ⊆ a ∧ (R “ a) ⊆ a) → x ∈ a))
29	26, 28	bitri 240	. . . . 5 ⊢ (x ∈ C ↔ ∀a((S ⊆ a ∧ (R “ a) ⊆ a) → x ∈ a))
30	25	eleq2i 2417	. . . . . 6 ⊢ (y ∈ C ↔ y ∈ ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)})
31		vex 2863	. . . . . . 7 ⊢ y ∈ V
32	31	elintab 3938	. . . . . 6 ⊢ (y ∈ ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)} ↔ ∀a((S ⊆ a ∧ (R “ a) ⊆ a) → y ∈ a))
33	30, 32	bitri 240	. . . . 5 ⊢ (y ∈ C ↔ ∀a((S ⊆ a ∧ (R “ a) ⊆ a) → y ∈ a))
34	22, 29, 33	3imtr4g 261	. . . 4 ⊢ (xRy → (x ∈ C → y ∈ C))
35	34	impcom 419	. . 3 ⊢ ((x ∈ C ∧ xRy) → y ∈ C)
36	6, 10, 35	vtocl2g 2919	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → ((A ∈ C ∧ ARB) → B ∈ C))
37	2, 36	mpcom 32	1 ⊢ ((A ∈ C ∧ ARB) → B ∈ C)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258  ∩cint 3927   class class class wbr 4640   “ cima 4723   Clos1 cclos1 5873

This proof depends on 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-br 4641  df-ima 4728  df-clos1 5874

This theorem is used by:  clos1induct  5881  clos1basesuc  5883  spaccl  6287  dmfrec  6317