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Theorem opthwiener 5169
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 4377 for other ordered pair definitions. (Contributed by NM, 28-Sep-2003.)
Hypotheses
Ref Expression
opthw.1 𝐴 ∈ V
opthw.2 𝐵 ∈ V
Assertion
Ref Expression
opthwiener ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opthwiener
StepHypRef Expression
1 id 22 . . . . . . 7 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
2 snex 5098 . . . . . . . . . . . 12 {{𝐵}} ∈ V
32prid2 4489 . . . . . . . . . . 11 {{𝐵}} ∈ {{{𝐴}, ∅}, {{𝐵}}}
4 eleq2 2874 . . . . . . . . . . 11 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → ({{𝐵}} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {{𝐵}} ∈ {{{𝐶}, ∅}, {{𝐷}}}))
53, 4mpbii 224 . . . . . . . . . 10 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{𝐵}} ∈ {{{𝐶}, ∅}, {{𝐷}}})
62elpr 4393 . . . . . . . . . 10 ({{𝐵}} ∈ {{{𝐶}, ∅}, {{𝐷}}} ↔ ({{𝐵}} = {{𝐶}, ∅} ∨ {{𝐵}} = {{𝐷}}))
75, 6sylib 209 . . . . . . . . 9 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → ({{𝐵}} = {{𝐶}, ∅} ∨ {{𝐵}} = {{𝐷}}))
8 0ex 4984 . . . . . . . . . . . . 13 ∅ ∈ V
98prid2 4489 . . . . . . . . . . . 12 ∅ ∈ {{𝐶}, ∅}
10 opthw.2 . . . . . . . . . . . . . 14 𝐵 ∈ V
1110snnz 4499 . . . . . . . . . . . . 13 {𝐵} ≠ ∅
128elsn 4385 . . . . . . . . . . . . . 14 (∅ ∈ {{𝐵}} ↔ ∅ = {𝐵})
13 eqcom 2813 . . . . . . . . . . . . . 14 (∅ = {𝐵} ↔ {𝐵} = ∅)
1412, 13bitri 266 . . . . . . . . . . . . 13 (∅ ∈ {{𝐵}} ↔ {𝐵} = ∅)
1511, 14nemtbir 3073 . . . . . . . . . . . 12 ¬ ∅ ∈ {{𝐵}}
16 nelneq2 2910 . . . . . . . . . . . 12 ((∅ ∈ {{𝐶}, ∅} ∧ ¬ ∅ ∈ {{𝐵}}) → ¬ {{𝐶}, ∅} = {{𝐵}})
179, 15, 16mp2an 675 . . . . . . . . . . 11 ¬ {{𝐶}, ∅} = {{𝐵}}
18 eqcom 2813 . . . . . . . . . . 11 ({{𝐶}, ∅} = {{𝐵}} ↔ {{𝐵}} = {{𝐶}, ∅})
1917, 18mtbi 313 . . . . . . . . . 10 ¬ {{𝐵}} = {{𝐶}, ∅}
20 biorf 951 . . . . . . . . . 10 (¬ {{𝐵}} = {{𝐶}, ∅} → ({{𝐵}} = {{𝐷}} ↔ ({{𝐵}} = {{𝐶}, ∅} ∨ {{𝐵}} = {{𝐷}})))
2119, 20ax-mp 5 . . . . . . . . 9 ({{𝐵}} = {{𝐷}} ↔ ({{𝐵}} = {{𝐶}, ∅} ∨ {{𝐵}} = {{𝐷}}))
227, 21sylibr 225 . . . . . . . 8 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{𝐵}} = {{𝐷}})
2322preq2d 4466 . . . . . . 7 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{{𝐶}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
241, 23eqtr4d 2843 . . . . . 6 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐵}}})
25 prex 5099 . . . . . . 7 {{𝐴}, ∅} ∈ V
26 prex 5099 . . . . . . 7 {{𝐶}, ∅} ∈ V
2725, 26preqr1 4567 . . . . . 6 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐵}}} → {{𝐴}, ∅} = {{𝐶}, ∅})
2824, 27syl 17 . . . . 5 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{𝐴}, ∅} = {{𝐶}, ∅})
29 snex 5098 . . . . . 6 {𝐴} ∈ V
30 snex 5098 . . . . . 6 {𝐶} ∈ V
3129, 30preqr1 4567 . . . . 5 ({{𝐴}, ∅} = {{𝐶}, ∅} → {𝐴} = {𝐶})
3228, 31syl 17 . . . 4 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {𝐴} = {𝐶})
33 opthw.1 . . . . 5 𝐴 ∈ V
3433sneqr 4559 . . . 4 ({𝐴} = {𝐶} → 𝐴 = 𝐶)
3532, 34syl 17 . . 3 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → 𝐴 = 𝐶)
36 snex 5098 . . . . . 6 {𝐵} ∈ V
3736sneqr 4559 . . . . 5 ({{𝐵}} = {{𝐷}} → {𝐵} = {𝐷})
3822, 37syl 17 . . . 4 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {𝐵} = {𝐷})
3910sneqr 4559 . . . 4 ({𝐵} = {𝐷} → 𝐵 = 𝐷)
4038, 39syl 17 . . 3 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → 𝐵 = 𝐷)
4135, 40jca 503 . 2 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → (𝐴 = 𝐶𝐵 = 𝐷))
42 sneq 4380 . . . . 5 (𝐴 = 𝐶 → {𝐴} = {𝐶})
4342preq1d 4465 . . . 4 (𝐴 = 𝐶 → {{𝐴}, ∅} = {{𝐶}, ∅})
4443preq1d 4465 . . 3 (𝐴 = 𝐶 → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐵}}})
45 sneq 4380 . . . . 5 (𝐵 = 𝐷 → {𝐵} = {𝐷})
46 sneq 4380 . . . . 5 ({𝐵} = {𝐷} → {{𝐵}} = {{𝐷}})
4745, 46syl 17 . . . 4 (𝐵 = 𝐷 → {{𝐵}} = {{𝐷}})
4847preq2d 4466 . . 3 (𝐵 = 𝐷 → {{{𝐶}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
4944, 48sylan9eq 2860 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
5041, 49impbii 200 1 ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384  wo 865   = wceq 1637  wcel 2156  Vcvv 3391  c0 4116  {csn 4370  {cpr 4372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-v 3393  df-dif 3772  df-un 3774  df-nul 4117  df-sn 4371  df-pr 4373
This theorem is referenced by: (None)
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