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Theorem opthwiener 4754
Description: Justification theorem for the ordered pair definition in Norbert Wiener, "A simplification of the logic of relations," Proc. of the Cambridge Philos. Soc., 1914, vol. 17, pp.387-390. It is also shown as a definition in [Enderton] p. 36 and as Exercise 4.8(b) of [Mendelson] p. 230. It is meaningful only for classes that exist as sets (i.e. are not proper classes). See df-op 4036 for other ordered pair definitions. (Contributed by NM, 28-Sep-2003.)
Hypotheses
Ref Expression
opthw.1
opthw.2
Assertion
Ref Expression
opthwiener

Proof of Theorem opthwiener
StepHypRef Expression
1 id 22 . . . . . . 7
2 snex 4693 . . . . . . . . . . . 12
32prid2 4139 . . . . . . . . . . 11
4 eleq2 2530 . . . . . . . . . . 11
53, 4mpbii 211 . . . . . . . . . 10
62elpr 4047 . . . . . . . . . 10
75, 6sylib 196 . . . . . . . . 9
8 0ex 4582 . . . . . . . . . . . . 13
98prid2 4139 . . . . . . . . . . . 12
10 opthw.2 . . . . . . . . . . . . . 14
1110snnz 4148 . . . . . . . . . . . . 13
128elsnc 4053 . . . . . . . . . . . . . 14
13 eqcom 2466 . . . . . . . . . . . . . 14
1412, 13bitri 249 . . . . . . . . . . . . 13
1511, 14nemtbir 2785 . . . . . . . . . . . 12
16 nelneq2 2575 . . . . . . . . . . . 12
179, 15, 16mp2an 672 . . . . . . . . . . 11
18 eqcom 2466 . . . . . . . . . . 11
1917, 18mtbi 298 . . . . . . . . . 10
20 biorf 405 . . . . . . . . . 10
2119, 20ax-mp 5 . . . . . . . . 9
227, 21sylibr 212 . . . . . . . 8
2322preq2d 4116 . . . . . . 7
241, 23eqtr4d 2501 . . . . . 6
25 prex 4694 . . . . . . 7
26 prex 4694 . . . . . . 7
2725, 26preqr1 4204 . . . . . 6
2824, 27syl 16 . . . . 5
29 snex 4693 . . . . . 6
30 snex 4693 . . . . . 6
3129, 30preqr1 4204 . . . . 5
3228, 31syl 16 . . . 4
33 opthw.1 . . . . 5
3433sneqr 4197 . . . 4
3532, 34syl 16 . . 3
36 snex 4693 . . . . . 6
3736sneqr 4197 . . . . 5
3822, 37syl 16 . . . 4
3910sneqr 4197 . . . 4
4038, 39syl 16 . . 3
4135, 40jca 532 . 2
42 sneq 4039 . . . . 5
4342preq1d 4115 . . . 4
4443preq1d 4115 . . 3
45 sneq 4039 . . . . 5
46 sneq 4039 . . . . 5
4745, 46syl 16 . . . 4
4847preq2d 4116 . . 3
4944, 48sylan9eq 2518 . 2
5041, 49impbii 188 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109   c0 3784  {csn 4029  {cpr 4031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-nul 3785  df-sn 4030  df-pr 4032
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