Theorem opthwiener 5428

Description: Justification theorem for the ordered pair definition in Norbert Wiener, A simplification of the logic of relations, Proceedings of the Cambridge Philosophical Society, 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 4568 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 5354	. . . . . . . . . . . 12 ⊢ {{𝐵}} ∈ V
3	2	prid2 4699	. . . . . . . . . . 11 ⊢ {{𝐵}} ∈ {{{𝐴}, ∅}, {{𝐵}}}
4		eleq2 2827	. . . . . . . . . . 11 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → ({{𝐵}} ∈ {{{𝐴}, ∅}, {{𝐵}}} ↔ {{𝐵}} ∈ {{{𝐶}, ∅}, {{𝐷}}}))
5	3, 4	mpbii 232	. . . . . . . . . 10 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{𝐵}} ∈ {{{𝐶}, ∅}, {{𝐷}}})
6	2	elpr 4584	. . . . . . . . . 10 ⊢ ({{𝐵}} ∈ {{{𝐶}, ∅}, {{𝐷}}} ↔ ({{𝐵}} = {{𝐶}, ∅} ∨ {{𝐵}} = {{𝐷}}))
7	5, 6	sylib 217	. . . . . . . . 9 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → ({{𝐵}} = {{𝐶}, ∅} ∨ {{𝐵}} = {{𝐷}}))
8		0ex 5231	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
9	8	prid2 4699	. . . . . . . . . . . 12 ⊢ ∅ ∈ {{𝐶}, ∅}
10		opthw.2	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
11	10	snnz 4712	. . . . . . . . . . . . 13 ⊢ {𝐵} ≠ ∅
12	8	elsn 4576	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ {{𝐵}} ↔ ∅ = {𝐵})
13		eqcom 2745	. . . . . . . . . . . . . 14 ⊢ (∅ = {𝐵} ↔ {𝐵} = ∅)
14	12, 13	bitri 274	. . . . . . . . . . . . 13 ⊢ (∅ ∈ {{𝐵}} ↔ {𝐵} = ∅)
15	11, 14	nemtbir 3040	. . . . . . . . . . . 12 ⊢ ¬ ∅ ∈ {{𝐵}}
16		nelneq2 2864	. . . . . . . . . . . 12 ⊢ ((∅ ∈ {{𝐶}, ∅} ∧ ¬ ∅ ∈ {{𝐵}}) → ¬ {{𝐶}, ∅} = {{𝐵}})
17	9, 15, 16	mp2an 689	. . . . . . . . . . 11 ⊢ ¬ {{𝐶}, ∅} = {{𝐵}}
18		eqcom 2745	. . . . . . . . . . 11 ⊢ ({{𝐶}, ∅} = {{𝐵}} ↔ {{𝐵}} = {{𝐶}, ∅})
19	17, 18	mtbi 322	. . . . . . . . . 10 ⊢ ¬ {{𝐵}} = {{𝐶}, ∅}
20		biorf 934	. . . . . . . . . 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 233	. . . . . . . 8 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{𝐵}} = {{𝐷}})
23	22	preq2d 4676	. . . . . . 7 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{{𝐶}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
24	1, 23	eqtr4d 2781	. . . . . 6 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐵}}})
25		prex 5355	. . . . . . 7 ⊢ {{𝐴}, ∅} ∈ V
26		prex 5355	. . . . . . 7 ⊢ {{𝐶}, ∅} ∈ V
27	25, 26	preqr1 4779	. . . . . 6 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐵}}} → {{𝐴}, ∅} = {{𝐶}, ∅})
28	24, 27	syl 17	. . . . 5 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {{𝐴}, ∅} = {{𝐶}, ∅})
29		snex 5354	. . . . . 6 ⊢ {𝐴} ∈ V
30		snex 5354	. . . . . 6 ⊢ {𝐶} ∈ V
31	29, 30	preqr1 4779	. . . . 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 4771	. . . 4 ⊢ ({𝐴} = {𝐶} → 𝐴 = 𝐶)
35	32, 34	syl 17	. . 3 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → 𝐴 = 𝐶)
36		snex 5354	. . . . . 6 ⊢ {𝐵} ∈ V
37	36	sneqr 4771	. . . . 5 ⊢ ({{𝐵}} = {{𝐷}} → {𝐵} = {𝐷})
38	22, 37	syl 17	. . . 4 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → {𝐵} = {𝐷})
39	10	sneqr 4771	. . . 4 ⊢ ({𝐵} = {𝐷} → 𝐵 = 𝐷)
40	38, 39	syl 17	. . 3 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → 𝐵 = 𝐷)
41	35, 40	jca 512	. 2 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
42		sneq 4571	. . . . 5 ⊢ (𝐴 = 𝐶 → {𝐴} = {𝐶})
43	42	preq1d 4675	. . . 4 ⊢ (𝐴 = 𝐶 → {{𝐴}, ∅} = {{𝐶}, ∅})
44	43	preq1d 4675	. . 3 ⊢ (𝐴 = 𝐶 → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐵}}})
45		sneq 4571	. . . . 5 ⊢ (𝐵 = 𝐷 → {𝐵} = {𝐷})
46		sneq 4571	. . . . 5 ⊢ ({𝐵} = {𝐷} → {{𝐵}} = {{𝐷}})
47	45, 46	syl 17	. . . 4 ⊢ (𝐵 = 𝐷 → {{𝐵}} = {{𝐷}})
48	47	preq2d 4676	. . 3 ⊢ (𝐵 = 𝐷 → {{{𝐶}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
49	44, 48	sylan9eq 2798	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}})
50	41, 49	impbii 208	1 ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  {csn 4561  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564

This theorem is referenced by: (None)