Theorem clos1basesuc 5883

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.

Step	Hyp	Ref	Expression
1		clos1basesuc.1	. . 3 ⊢ S ∈ V
2		clos1basesuc.2	. . 3 ⊢ R ∈ V
3		clos1basesuc.3	. . 3 ⊢ C = Clos1 (S, R)
4		abid2 2471	. . . . . . 7 ⊢ {y ∣ y ∈ S} = S
5	4	eqcomi 2357	. . . . . 6 ⊢ S = {y ∣ y ∈ S}
6		df-ima 4728	. . . . . 6 ⊢ (R “ C) = {y ∣ ∃x ∈ C xRy}
7	5, 6	uneq12i 3417	. . . . 5 ⊢ (S ∪ (R “ C)) = ({y ∣ y ∈ S} ∪ {y ∣ ∃x ∈ C xRy})
8		unab 3522	. . . . 5 ⊢ ({y ∣ y ∈ S} ∪ {y ∣ ∃x ∈ C xRy}) = {y ∣ (y ∈ S ∨ ∃x ∈ C xRy)}
9	7, 8	eqtri 2373	. . . 4 ⊢ (S ∪ (R “ C)) = {y ∣ (y ∈ S ∨ ∃x ∈ C xRy)}
10	1, 2	clos1ex 5877	. . . . . . 7 ⊢ Clos1 (S, R) ∈ V
11	3, 10	eqeltri 2423	. . . . . 6 ⊢ C ∈ V
12	2, 11	imaex 4748	. . . . 5 ⊢ (R “ C) ∈ V
13	1, 12	unex 4107	. . . 4 ⊢ (S ∪ (R “ C)) ∈ V
14	9, 13	eqeltrri 2424	. . 3 ⊢ {y ∣ (y ∈ S ∨ ∃x ∈ C xRy)} ∈ V
15		eleq1 2413	. . . 4 ⊢ (y = z → (y ∈ S ↔ z ∈ S))
16		breq2 4644	. . . . 5 ⊢ (y = z → (xRy ↔ xRz))
17	16	rexbidv 2636	. . . 4 ⊢ (y = z → (∃x ∈ C xRy ↔ ∃x ∈ C xRz))
18	15, 17	orbi12d 690	. . 3 ⊢ (y = z → ((y ∈ S ∨ ∃x ∈ C xRy) ↔ (z ∈ S ∨ ∃x ∈ C xRz)))
19		eleq1 2413	. . . 4 ⊢ (y = w → (y ∈ S ↔ w ∈ S))
20		breq2 4644	. . . . 5 ⊢ (y = w → (xRy ↔ xRw))
21	20	rexbidv 2636	. . . 4 ⊢ (y = w → (∃x ∈ C xRy ↔ ∃x ∈ C xRw))
22	19, 21	orbi12d 690	. . 3 ⊢ (y = w → ((y ∈ S ∨ ∃x ∈ C xRy) ↔ (w ∈ S ∨ ∃x ∈ C xRw)))
23		eleq1 2413	. . . 4 ⊢ (y = A → (y ∈ S ↔ A ∈ S))
24		breq2 4644	. . . . 5 ⊢ (y = A → (xRy ↔ xRA))
25	24	rexbidv 2636	. . . 4 ⊢ (y = A → (∃x ∈ C xRy ↔ ∃x ∈ C xRA))
26	23, 25	orbi12d 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))
28	3	clos1base 5879	. . . . . . . . . 10 ⊢ S ⊆ C
29	28	sseli 3270	. . . . . . . . 9 ⊢ (z ∈ S → z ∈ C)
30		breq1 4643	. . . . . . . . . . 11 ⊢ (y = z → (yRw ↔ zRw))
31	30	rspcev 2956	. . . . . . . . . 10 ⊢ ((z ∈ C ∧ zRw) → ∃y ∈ C yRw)
32	31	ex 423	. . . . . . . . 9 ⊢ (z ∈ C → (zRw → ∃y ∈ C yRw))
33	29, 32	syl 15	. . . . . . . 8 ⊢ (z ∈ S → (zRw → ∃y ∈ C yRw))
34	3	clos1conn 5880	. . . . . . . . . 10 ⊢ ((x ∈ C ∧ xRz) → z ∈ C)
35	34, 32	syl 15	. . . . . . . . 9 ⊢ ((x ∈ C ∧ xRz) → (zRw → ∃y ∈ C yRw))
36	35	rexlimiva 2734	. . . . . . . 8 ⊢ (∃x ∈ C xRz → (zRw → ∃y ∈ C yRw))
37	33, 36	jaoi 368	. . . . . . 7 ⊢ ((z ∈ S ∨ ∃x ∈ C xRz) → (zRw → ∃y ∈ C yRw))
38	37	impcom 419	. . . . . 6 ⊢ ((zRw ∧ (z ∈ S ∨ ∃x ∈ C xRz)) → ∃y ∈ C yRw)
39		breq1 4643	. . . . . . 7 ⊢ (x = y → (xRw ↔ yRw))
40	39	cbvrexv 2837	. . . . . 6 ⊢ (∃x ∈ C xRw ↔ ∃y ∈ C yRw)
41	38, 40	sylibr 203	. . . . 5 ⊢ ((zRw ∧ (z ∈ S ∨ ∃x ∈ C xRz)) → ∃x ∈ C xRw)
42	41	olcd 382	. . . 4 ⊢ ((zRw ∧ (z ∈ S ∨ ∃x ∈ C xRz)) → (w ∈ S ∨ ∃x ∈ C xRw))
43	42	3adant1 973	. . 3 ⊢ ((z ∈ C ∧ zRw ∧ (z ∈ S ∨ ∃x ∈ C xRz)) → (w ∈ S ∨ ∃x ∈ C xRw))
44	1, 2, 3, 14, 18, 22, 26, 27, 43	clos1is 5882	. 2 ⊢ (A ∈ C → (A ∈ S ∨ ∃x ∈ C xRA))
45	28	sseli 3270	. . 3 ⊢ (A ∈ S → A ∈ C)
46	3	clos1conn 5880	. . . 4 ⊢ ((x ∈ C ∧ xRA) → A ∈ C)
47	46	rexlimiva 2734	. . 3 ⊢ (∃x ∈ C xRA → A ∈ C)
48	45, 47	jaoi 368	. 2 ⊢ ((A ∈ S ∨ ∃x ∈ C xRA) → A ∈ C)
49	44, 48	impbii 180	1 ⊢ (A ∈ C ↔ (A ∈ S ∨ ∃x ∈ C xRA))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ∪ cun 3208   class class class wbr 4640   “ cima 4723   Clos1 cclos1 5873

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 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 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 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-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-2nd 4798  df-txp 5737  df-fix 5741  df-ins2 5751  df-ins3 5753  df-image 5755  df-clos1 5874

This theorem is referenced by:  clos1baseima  5884  clos1basesucg  5885