ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssiinf Structured version   GIF version

Theorem ssiinf 3676
Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
ssiinf.1 x𝐶
Assertion
Ref Expression
ssiinf (𝐶 x A Bx A 𝐶B)

Proof of Theorem ssiinf
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 vex 2534 . . . . 5 y V
2 eliin 3632 . . . . 5 (y V → (y x A Bx A y B))
31, 2ax-mp 7 . . . 4 (y x A Bx A y B)
43ralbii 2304 . . 3 (y 𝐶 y x A By 𝐶 x A y B)
5 ssiinf.1 . . . 4 x𝐶
6 nfcv 2156 . . . 4 yA
75, 6ralcomf 2445 . . 3 (y 𝐶 x A y Bx A y 𝐶 y B)
84, 7bitri 173 . 2 (y 𝐶 y x A Bx A y 𝐶 y B)
9 dfss3 2908 . 2 (𝐶 x A By 𝐶 y x A B)
10 dfss3 2908 . . 3 (𝐶By 𝐶 y B)
1110ralbii 2304 . 2 (x A 𝐶Bx A y 𝐶 y B)
128, 9, 113bitr4i 201 1 (𝐶 x A Bx A 𝐶B)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1370  wnfc 2143  wral 2280  Vcvv 2531  wss 2890   ciin 3628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533  df-in 2897  df-ss 2904  df-iin 3630
This theorem is referenced by:  ssiin  3677  dmiin  4503
  Copyright terms: Public domain W3C validator