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 ∈ V

Assertion

Ref	Expression
brsnsi1	⊢ ({A} SI RB ↔ ∃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 ↔ ∃y∃x({A} = {y} ∧ B = {x} ∧ yRx))
2		excom 1741	. . 3 ⊢ (∃y∃x({A} = {y} ∧ B = {x} ∧ yRx) ↔ ∃x∃y({A} = {y} ∧ B = {x} ∧ yRx))
3		eqcom 2355	. . . . . . . . 9 ⊢ ({A} = {y} ↔ {y} = {A})
4		vex 2863	. . . . . . . . . 10 ⊢ y ∈ 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 ⊢ (∃y({A} = {y} ∧ B = {x} ∧ yRx) ↔ ∃y(y = A ∧ (B = {x} ∧ yRx)))
11		brsnsi1.1	. . . . . 6 ⊢ A ∈ 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 ⊢ (∃y(y = A ∧ (B = {x} ∧ yRx)) ↔ (B = {x} ∧ ARx))
15	10, 14	bitri 240	. . . 4 ⊢ (∃y({A} = {y} ∧ B = {x} ∧ yRx) ↔ (B = {x} ∧ ARx))
16	15	exbii 1582	. . 3 ⊢ (∃x∃y({A} = {y} ∧ B = {x} ∧ yRx) ↔ ∃x(B = {x} ∧ ARx))
17	2, 16	bitri 240	. 2 ⊢ (∃y∃x({A} = {y} ∧ B = {x} ∧ yRx) ↔ ∃x(B = {x} ∧ ARx))
18	1, 17	bitri 240	1 ⊢ ({A} SI RB ↔ ∃x(B = {x} ∧ ARx))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ 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