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Theorem brsnsi1 5775
 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 RBx(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.
StepHypRef Expression
1 brsi 4761 . 2 ({A} SI RByx({A} = {y} B = {x} yRx))
2 excom 1741 . . 3 (yx({A} = {y} B = {x} yRx) ↔ xy({A} = {y} B = {x} yRx))
3 eqcom 2355 . . . . . . . . 9 ({A} = {y} ↔ {y} = {A})
4 vex 2862 . . . . . . . . . 10 y V
54sneqb 3876 . . . . . . . . 9 ({y} = {A} ↔ y = A)
63, 5bitri 240 . . . . . . . 8 ({A} = {y} ↔ y = A)
763anbi1i 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)))
97, 8bitri 240 . . . . . 6 (({A} = {y} B = {x} yRx) ↔ (y = A (B = {x} yRx)))
109exbii 1582 . . . . 5 (y({A} = {y} B = {x} yRx) ↔ y(y = A (B = {x} yRx)))
11 brsnsi1.1 . . . . . 6 A V
12 breq1 4642 . . . . . . 7 (y = A → (yRxARx))
1312anbi2d 684 . . . . . 6 (y = A → ((B = {x} yRx) ↔ (B = {x} ARx)))
1411, 13ceqsexv 2894 . . . . 5 (y(y = A (B = {x} yRx)) ↔ (B = {x} ARx))
1510, 14bitri 240 . . . 4 (y({A} = {y} B = {x} yRx) ↔ (B = {x} ARx))
1615exbii 1582 . . 3 (xy({A} = {y} B = {x} yRx) ↔ x(B = {x} ARx))
172, 16bitri 240 . 2 (yx({A} = {y} B = {x} yRx) ↔ x(B = {x} ARx))
181, 17bitri 240 1 ({A} SI RBx(B = {x} ARx))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ 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-1 5  ax-2 6  ax-3 7  ax-mp 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:  ceexlem1  6173  tcfnex  6244  nchoicelem11  6299  nchoicelem16  6304
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