Theorem connexd 5932

Description: The connectivity property. (Contributed by SF, 18-Mar-2015.)

Hypotheses

Ref	Expression
connexd.1	|- (ph -> R Connex A)
connexd.2	|- (ph -> X e. A)
connexd.3	|- (ph -> Y e. A)

Assertion

Ref	Expression
connexd	|- (ph -> (XRY \/ YRX))

Proof of Theorem connexd

Dummy variables a r x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		connexd.1	. 2 |- (ph -> R Connex A)
2		brex 4690	. . . . 5 |- (R Connex A -> (R e. _V /\ A e. _V))
3		breq 4642	. . . . . . . 8 |- (r = R -> (xry <-> xRy))
4		breq 4642	. . . . . . . 8 |- (r = R -> (yrx <-> yRx))
5	3, 4	orbi12d 690	. . . . . . 7 |- (r = R -> ((xry \/ yrx) <-> (xRy \/ yRx)))
6	5	2ralbidv 2657	. . . . . 6 |- (r = R -> (A.x e. a A.y e. a (xry \/ yrx) <-> A.x e. a A.y e. a (xRy \/ yRx)))
7		raleq 2808	. . . . . . 7 |- (a = A -> (A.y e. a (xRy \/ yRx) <-> A.y e. A (xRy \/ yRx)))
8	7	raleqbi1dv 2816	. . . . . 6 |- (a = A -> (A.x e. a A.y e. a (xRy \/ yRx) <-> A.x e. A A.y e. A (xRy \/ yRx)))
9		df-connex 5904	. . . . . 6 |- Connex = {<.r, a>. | A.x e. a A.y e. a (xry \/ yrx)}
10	6, 8, 9	brabg 4707	. . . . 5 |- ((R e. _V /\ A e. _V) -> (R Connex A <-> A.x e. A A.y e. A (xRy \/ yRx)))
11	2, 10	syl 15	. . . 4 |- (R Connex A -> (R Connex A <-> A.x e. A A.y e. A (xRy \/ yRx)))
12	11	ibi 232	. . 3 |- (R Connex A -> A.x e. A A.y e. A (xRy \/ yRx))
13		connexd.2	. . . 4 |- (ph -> X e. A)
14		connexd.3	. . . 4 |- (ph -> Y e. A)
15		breq1 4643	. . . . . 6 |- (x = X -> (xRy <-> XRy))
16		breq2 4644	. . . . . 6 |- (x = X -> (yRx <-> yRX))
17	15, 16	orbi12d 690	. . . . 5 |- (x = X -> ((xRy \/ yRx) <-> (XRy \/ yRX)))
18		breq2 4644	. . . . . 6 |- (y = Y -> (XRy <-> XRY))
19		breq1 4643	. . . . . 6 |- (y = Y -> (yRX <-> YRX))
20	18, 19	orbi12d 690	. . . . 5 |- (y = Y -> ((XRy \/ yRX) <-> (XRY \/ YRX)))
21	17, 20	rspc2v 2962	. . . 4 |- ((X e. A /\ Y e. A) -> (A.x e. A A.y e. A (xRy \/ yRx) -> (XRY \/ YRX)))
22	13, 14, 21	syl2anc 642	. . 3 |- (ph -> (A.x e. A A.y e. A (xRy \/ yRx) -> (XRY \/ YRX)))
23	12, 22	syl5 28	. 2 |- (ph -> (R Connex A -> (XRY \/ YRX)))
24	1, 23	mpd 14	1 |- (ph -> (XRY \/ YRX))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2615  _Vcvv 2860   class class class wbr 4640   Connex cconnex 5893

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-connex 5904

This theorem is referenced by:  weds  5939  nchoicelem8  6297