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Theorem difprsnss 3502
 Description: Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
difprsnss

Proof of Theorem difprsnss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . . 5
21elpr 3396 . . . 4
3 velsn 3392 . . . . 5
43notbii 594 . . . 4
5 biorf 663 . . . . 5
65biimparc 283 . . . 4
72, 4, 6syl2anb 275 . . 3
8 eldif 2927 . . 3
9 velsn 3392 . . 3
107, 8, 93imtr4i 190 . 2
1110ssriv 2949 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 97   wo 629   wceq 1243   wcel 1393   cdif 2914   wss 2917  csn 3375  cpr 3376 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-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382 This theorem is referenced by: (None)
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