Theorem opthwiener 5403

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 4573 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 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem opthwiener

Step	Hyp	Ref	Expression
1		id 22	. . . . . . 7 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
2		snex 5331	. . . . . . . . . . . 12 ⊢ {{𝐵}} ∈ V
3	2	prid2 4698	. . . . . . . . . . 11 ⊢ {{𝐵}} ∈ {{{𝐴}, ∅}, {{𝐵}}}
4		eleq2 2901	. . . . . . . . . . 11 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → ({{𝐵}} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {{𝐵}} ∈ {{{𝐶}, ∅}, {{𝐷}}}))
5	3, 4	mpbii 235	. . . . . . . . . 10 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{𝐵}} ∈ {{{𝐶}, ∅}, {{𝐷}}})
6	2	elpr 4589	. . . . . . . . . 10 ⊢ ({{𝐵}} ∈ {{{𝐶}, ∅}, {{𝐷}}} ↔ ({{𝐵}} = {{𝐶}, ∅} ∨ {{𝐵}} = {{𝐷}}))
7	5, 6	sylib 220	. . . . . . . . 9 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → ({{𝐵}} = {{𝐶}, ∅} ∨ {{𝐵}} = {{𝐷}}))
8		0ex 5210	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
9	8	prid2 4698	. . . . . . . . . . . 12 ⊢ ∅ ∈ {{𝐶}, ∅}
10		opthw.2	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
11	10	snnz 4710	. . . . . . . . . . . . 13 ⊢ {𝐵} ≠ ∅
12	8	elsn 4581	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ {{𝐵}} ↔ ∅ = {𝐵})
13		eqcom 2828	. . . . . . . . . . . . . 14 ⊢ (∅ = {𝐵} ↔ {𝐵} = ∅)
14	12, 13	bitri 277	. . . . . . . . . . . . 13 ⊢ (∅ ∈ {{𝐵}} ↔ {𝐵} = ∅)
15	11, 14	nemtbir 3112	. . . . . . . . . . . 12 ⊢ ¬ ∅ ∈ {{𝐵}}
16		nelneq2 2938	. . . . . . . . . . . 12 ⊢ ((∅ ∈ {{𝐶}, ∅} ∧ ¬ ∅ ∈ {{𝐵}}) → ¬ {{𝐶}, ∅} = {{𝐵}})
17	9, 15, 16	mp2an 690	. . . . . . . . . . 11 ⊢ ¬ {{𝐶}, ∅} = {{𝐵}}
18		eqcom 2828	. . . . . . . . . . 11 ⊢ ({{𝐶}, ∅} = {{𝐵}} ↔ {{𝐵}} = {{𝐶}, ∅})
19	17, 18	mtbi 324	. . . . . . . . . 10 ⊢ ¬ {{𝐵}} = {{𝐶}, ∅}
20		biorf 933	. . . . . . . . . 10 ⊢ (¬ {{𝐵}} = {{𝐶}, ∅} → ({{𝐵}} = {{𝐷}} ↔ ({{𝐵}} = {{𝐶}, ∅} ∨ {{𝐵}} = {{𝐷}})))
21	19, 20	ax-mp 5	. . . . . . . . 9 ⊢ ({{𝐵}} = {{𝐷}} ↔ ({{𝐵}} = {{𝐶}, ∅} ∨ {{𝐵}} = {{𝐷}}))
22	7, 21	sylibr 236	. . . . . . . 8 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{𝐵}} = {{𝐷}})
23	22	preq2d 4675	. . . . . . 7 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{{𝐶}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
24	1, 23	eqtr4d 2859	. . . . . 6 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐵}}})
25		prex 5332	. . . . . . 7 ⊢ {{𝐴}, ∅} ∈ V
26		prex 5332	. . . . . . 7 ⊢ {{𝐶}, ∅} ∈ V
27	25, 26	preqr1 4778	. . . . . 6 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐵}}} → {{𝐴}, ∅} = {{𝐶}, ∅})
28	24, 27	syl 17	. . . . 5 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{𝐴}, ∅} = {{𝐶}, ∅})
29		snex 5331	. . . . . 6 ⊢ {𝐴} ∈ V
30		snex 5331	. . . . . 6 ⊢ {𝐶} ∈ V
31	29, 30	preqr1 4778	. . . . 5 ⊢ ({{𝐴}, ∅} = {{𝐶}, ∅} → {𝐴} = {𝐶})
32	28, 31	syl 17	. . . 4 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {𝐴} = {𝐶})
33		opthw.1	. . . . 5 ⊢ 𝐴 ∈ V
34	33	sneqr 4770	. . . 4 ⊢ ({𝐴} = {𝐶} → 𝐴 = 𝐶)
35	32, 34	syl 17	. . 3 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → 𝐴 = 𝐶)
36		snex 5331	. . . . . 6 ⊢ {𝐵} ∈ V
37	36	sneqr 4770	. . . . 5 ⊢ ({{𝐵}} = {{𝐷}} → {𝐵} = {𝐷})
38	22, 37	syl 17	. . . 4 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {𝐵} = {𝐷})
39	10	sneqr 4770	. . . 4 ⊢ ({𝐵} = {𝐷} → 𝐵 = 𝐷)
40	38, 39	syl 17	. . 3 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → 𝐵 = 𝐷)
41	35, 40	jca 514	. 2 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
42		sneq 4576	. . . . 5 ⊢ (𝐴 = 𝐶 → {𝐴} = {𝐶})
43	42	preq1d 4674	. . . 4 ⊢ (𝐴 = 𝐶 → {{𝐴}, ∅} = {{𝐶}, ∅})
44	43	preq1d 4674	. . 3 ⊢ (𝐴 = 𝐶 → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐵}}})
45		sneq 4576	. . . . 5 ⊢ (𝐵 = 𝐷 → {𝐵} = {𝐷})
46		sneq 4576	. . . . 5 ⊢ ({𝐵} = {𝐷} → {{𝐵}} = {{𝐷}})
47	45, 46	syl 17	. . . 4 ⊢ (𝐵 = 𝐷 → {{𝐵}} = {{𝐷}})
48	47	preq2d 4675	. . 3 ⊢ (𝐵 = 𝐷 → {{{𝐶}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
49	44, 48	sylan9eq 2876	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
50	41, 49	impbii 211	1 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ∅c0 4290  {csn 4566  {cpr 4568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4567  df-pr 4569

This theorem is referenced by: (None)