Theorem brsnsi 5774

Description: Binary relationship of singletons in a singleton image. (Contributed by SF, 9-Feb-2015.)

Hypotheses

Ref	Expression
brsnsi.1	⊢ A ∈ V
brsnsi.2	⊢ B ∈ V

Assertion

Ref	Expression
brsnsi	⊢ ({A} SI R{B} ↔ ARB)

Proof of Theorem brsnsi

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4112	. . 3 ⊢ {A} ∈ V
2		snex 4112	. . 3 ⊢ {B} ∈ V
3		eqeq1 2359	. . . . . 6 ⊢ (z = {A} → (z = {x} ↔ {A} = {x}))
4		eqcom 2355	. . . . . . 7 ⊢ ({A} = {x} ↔ {x} = {A})
5		vex 2863	. . . . . . . 8 ⊢ x ∈ V
6	5	sneqb 3877	. . . . . . 7 ⊢ ({x} = {A} ↔ x = A)
7	4, 6	bitri 240	. . . . . 6 ⊢ ({A} = {x} ↔ x = A)
8	3, 7	syl6bb 252	. . . . 5 ⊢ (z = {A} → (z = {x} ↔ x = A))
9	8	3anbi1d 1256	. . . 4 ⊢ (z = {A} → ((z = {x} ∧ w = {y} ∧ xRy) ↔ (x = A ∧ w = {y} ∧ xRy)))
10	9	2exbidv 1628	. . 3 ⊢ (z = {A} → (∃x∃y(z = {x} ∧ w = {y} ∧ xRy) ↔ ∃x∃y(x = A ∧ w = {y} ∧ xRy)))
11		eqeq1 2359	. . . . . 6 ⊢ (w = {B} → (w = {y} ↔ {B} = {y}))
12		eqcom 2355	. . . . . . 7 ⊢ ({B} = {y} ↔ {y} = {B})
13		vex 2863	. . . . . . . 8 ⊢ y ∈ V
14	13	sneqb 3877	. . . . . . 7 ⊢ ({y} = {B} ↔ y = B)
15	12, 14	bitri 240	. . . . . 6 ⊢ ({B} = {y} ↔ y = B)
16	11, 15	syl6bb 252	. . . . 5 ⊢ (w = {B} → (w = {y} ↔ y = B))
17	16	3anbi2d 1257	. . . 4 ⊢ (w = {B} → ((x = A ∧ w = {y} ∧ xRy) ↔ (x = A ∧ y = B ∧ xRy)))
18	17	2exbidv 1628	. . 3 ⊢ (w = {B} → (∃x∃y(x = A ∧ w = {y} ∧ xRy) ↔ ∃x∃y(x = A ∧ y = B ∧ xRy)))
19		df-si 4729	. . 3 ⊢ SI R = {⟨z, w⟩ ∣ ∃x∃y(z = {x} ∧ w = {y} ∧ xRy)}
20	1, 2, 10, 18, 19	brab 4710	. 2 ⊢ ({A} SI R{B} ↔ ∃x∃y(x = A ∧ y = B ∧ xRy))
21		brsnsi.1	. . 3 ⊢ A ∈ V
22		brsnsi.2	. . 3 ⊢ B ∈ V
23		breq1 4643	. . 3 ⊢ (x = A → (xRy ↔ ARy))
24		breq2 4644	. . 3 ⊢ (y = B → (ARy ↔ ARB))
25	21, 22, 23, 24	ceqsex2v 2897	. 2 ⊢ (∃x∃y(x = A ∧ y = B ∧ xRy) ↔ ARB)
26	20, 25	bitri 240	1 ⊢ ({A} SI R{B} ↔ ARB)

Syntax hints:   ↔ wb 176   ∧ 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:  opsnelsi  5775  pw1fnex  5853  tcfnex  6245