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Theorem brsnsi 5773
 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.
StepHypRef Expression
1 snex 4111 . . 3 {A} V
2 snex 4111 . . 3 {B} V
3 eqeq1 2359 . . . . . 6 (z = {A} → (z = {x} ↔ {A} = {x}))
4 eqcom 2355 . . . . . . 7 ({A} = {x} ↔ {x} = {A})
5 vex 2862 . . . . . . . 8 x V
65sneqb 3876 . . . . . . 7 ({x} = {A} ↔ x = A)
74, 6bitri 240 . . . . . 6 ({A} = {x} ↔ x = A)
83, 7syl6bb 252 . . . . 5 (z = {A} → (z = {x} ↔ x = A))
983anbi1d 1256 . . . 4 (z = {A} → ((z = {x} w = {y} xRy) ↔ (x = A w = {y} xRy)))
1092exbidv 1628 . . 3 (z = {A} → (xy(z = {x} w = {y} xRy) ↔ xy(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 2862 . . . . . . . 8 y V
1413sneqb 3876 . . . . . . 7 ({y} = {B} ↔ y = B)
1512, 14bitri 240 . . . . . 6 ({B} = {y} ↔ y = B)
1611, 15syl6bb 252 . . . . 5 (w = {B} → (w = {y} ↔ y = B))
17163anbi2d 1257 . . . 4 (w = {B} → ((x = A w = {y} xRy) ↔ (x = A y = B xRy)))
18172exbidv 1628 . . 3 (w = {B} → (xy(x = A w = {y} xRy) ↔ xy(x = A y = B xRy)))
19 df-si 4728 . . 3 SI R = {z, w xy(z = {x} w = {y} xRy)}
201, 2, 10, 18, 19brab 4709 . 2 ({A} SI R{B} ↔ xy(x = A y = B xRy))
21 brsnsi.1 . . 3 A V
22 brsnsi.2 . . 3 B V
23 breq1 4642 . . 3 (x = A → (xRyARy))
24 breq2 4643 . . 3 (y = B → (ARyARB))
2521, 22, 23, 24ceqsex2v 2896 . 2 (xy(x = A y = B xRy) ↔ ARB)
2620, 25bitri 240 1 ({A} SI R{B} ↔ ARB)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737   class class class wbr 4639   SI csi 4720 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-si 4728 This theorem is referenced by:  opsnelsi  5774  pw1fnex  5852  tcfnex  6244
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