Theorem brsnsi1 5776

Description: Binary relationship of a singleton to an arbitrary set in a singleton image. (Contributed by SF, 9-Mar-2015.)

Hypothesis

Ref	Expression
brsnsi1.1	|- A e. _V

Assertion

Ref	Expression
brsnsi1	|- ({A} SI RB <-> E.x(B = {x} /\ ARx))

Distinct variable groups:   x,A   x,B   x,R

Proof of Theorem brsnsi1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brsi 4762	. 2 |- ({A} SI RB <-> E.yE.x({A} = {y} /\ B = {x} /\ yRx))
2		excom 1741	. . 3 |- (E.yE.x({A} = {y} /\ B = {x} /\ yRx) <-> E.xE.y({A} = {y} /\ B = {x} /\ yRx))
3		eqcom 2355	. . . . . . . . 9 |- ({A} = {y} <-> {y} = {A})
4		vex 2863	. . . . . . . . . 10 |- y e. _V
5	4	sneqb 3877	. . . . . . . . 9 |- ({y} = {A} <-> y = A)
6	3, 5	bitri 240	. . . . . . . 8 |- ({A} = {y} <-> y = A)
7	6	3anbi1i 1142	. . . . . . 7 |- (({A} = {y} /\ B = {x} /\ yRx) <-> (y = A /\ B = {x} /\ yRx))
8		3anass 938	. . . . . . 7 |- ((y = A /\ B = {x} /\ yRx) <-> (y = A /\ (B = {x} /\ yRx)))
9	7, 8	bitri 240	. . . . . 6 |- (({A} = {y} /\ B = {x} /\ yRx) <-> (y = A /\ (B = {x} /\ yRx)))
10	9	exbii 1582	. . . . 5 |- (E.y({A} = {y} /\ B = {x} /\ yRx) <-> E.y(y = A /\ (B = {x} /\ yRx)))
11		brsnsi1.1	. . . . . 6 |- A e. _V
12		breq1 4643	. . . . . . 7 |- (y = A -> (yRx <-> ARx))
13	12	anbi2d 684	. . . . . 6 |- (y = A -> ((B = {x} /\ yRx) <-> (B = {x} /\ ARx)))
14	11, 13	ceqsexv 2895	. . . . 5 |- (E.y(y = A /\ (B = {x} /\ yRx)) <-> (B = {x} /\ ARx))
15	10, 14	bitri 240	. . . 4 |- (E.y({A} = {y} /\ B = {x} /\ yRx) <-> (B = {x} /\ ARx))
16	15	exbii 1582	. . 3 |- (E.xE.y({A} = {y} /\ B = {x} /\ yRx) <-> E.x(B = {x} /\ ARx))
17	2, 16	bitri 240	. 2 |- (E.yE.x({A} = {y} /\ B = {x} /\ yRx) <-> E.x(B = {x} /\ ARx))
18	1, 17	bitri 240	1 |- ({A} SI RB <-> E.x(B = {x} /\ ARx))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860  {csn 3738   class class class wbr 4640   SI csi 4721

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-si 4729

This theorem is referenced by:  ceexlem1  6174  tcfnex  6245  nchoicelem11  6300  nchoicelem16  6305