Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  opthwiener Unicode version

Theorem opthwiener 4445
 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 3810 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 20 . . . . . . 7
2 snex 4392 . . . . . . . . . . . 12
32prid2 3900 . . . . . . . . . . 11
4 eleq2 2491 . . . . . . . . . . 11
53, 4mpbii 203 . . . . . . . . . 10
62elpr 3819 . . . . . . . . . 10
75, 6sylib 189 . . . . . . . . 9
8 0ex 4326 . . . . . . . . . . . . 13
98prid2 3900 . . . . . . . . . . . 12
10 opthw.2 . . . . . . . . . . . . . 14
1110snnz 3909 . . . . . . . . . . . . 13
128elsnc 3824 . . . . . . . . . . . . . 14
13 eqcom 2432 . . . . . . . . . . . . . 14
1412, 13bitri 241 . . . . . . . . . . . . 13
1511, 14nemtbir 2681 . . . . . . . . . . . 12
16 nelneq2 2529 . . . . . . . . . . . 12
179, 15, 16mp2an 654 . . . . . . . . . . 11
18 eqcom 2432 . . . . . . . . . . 11
1917, 18mtbi 290 . . . . . . . . . 10
20 biorf 395 . . . . . . . . . 10
2119, 20ax-mp 8 . . . . . . . . 9
227, 21sylibr 204 . . . . . . . 8
2322preq2d 3877 . . . . . . 7
241, 23eqtr4d 2465 . . . . . 6
25 prex 4393 . . . . . . 7
26 prex 4393 . . . . . . 7
2725, 26preqr1 3959 . . . . . 6
2824, 27syl 16 . . . . 5
29 snex 4392 . . . . . 6
30 snex 4392 . . . . . 6
3129, 30preqr1 3959 . . . . 5
3228, 31syl 16 . . . 4
33 opthw.1 . . . . 5
3433sneqr 3953 . . . 4
3532, 34syl 16 . . 3
36 snex 4392 . . . . . 6
3736sneqr 3953 . . . . 5
3822, 37syl 16 . . . 4
3910sneqr 3953 . . . 4
4038, 39syl 16 . . 3
4135, 40jca 519 . 2
42 sneq 3812 . . . . 5
4342preq1d 3876 . . . 4
4443preq1d 3876 . . 3
45 sneq 3812 . . . . 5
46 sneq 3812 . . . . 5
4745, 46syl 16 . . . 4
4847preq2d 3877 . . 3
4944, 48sylan9eq 2482 . 2
5041, 49impbii 181 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 177   wo 358   wa 359   wceq 1652   wcel 1725  cvv 2943  c0 3615  csn 3801  cpr 3802 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-v 2945  df-dif 3310  df-un 3312  df-nul 3616  df-sn 3807  df-pr 3808
 Copyright terms: Public domain W3C validator