Theorem preq12b 4855

Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Hypotheses

Ref	Expression
preqr1.a	⊢ 𝐴 ∈ V
preqr1.b	⊢ 𝐵 ∈ V
preq12b.c	⊢ 𝐶 ∈ V
preq12b.d	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
preq12b	⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))

Proof of Theorem preq12b

Step	Hyp	Ref	Expression
1		preqr1.a	. . . . . 6 ⊢ 𝐴 ∈ V
2	1	prid1 4767	. . . . 5 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		eleq2 2828	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
4	2, 3	mpbii 233	. . . 4 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
5	1	elpr 4655	. . . 4 ⊢ (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
6	4, 5	sylib 218	. . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
7		preq1 4738	. . . . . . . 8 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
8	7	eqeq1d 2737	. . . . . . 7 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
9		preqr1.b	. . . . . . . 8 ⊢ 𝐵 ∈ V
10		preq12b.d	. . . . . . . 8 ⊢ 𝐷 ∈ V
11	9, 10	preqr2 4854	. . . . . . 7 ⊢ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
12	8, 11	biimtrdi 253	. . . . . 6 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
13	12	com12 32	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 → 𝐵 = 𝐷))
14	13	ancld 550	. . . 4 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
15		prcom 4737	. . . . . . 7 ⊢ {𝐶, 𝐷} = {𝐷, 𝐶}
16	15	eqeq2i 2748	. . . . . 6 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
17		preq1 4738	. . . . . . . . 9 ⊢ (𝐴 = 𝐷 → {𝐴, 𝐵} = {𝐷, 𝐵})
18	17	eqeq1d 2737	. . . . . . . 8 ⊢ (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} ↔ {𝐷, 𝐵} = {𝐷, 𝐶}))
19		preq12b.c	. . . . . . . . 9 ⊢ 𝐶 ∈ V
20	9, 19	preqr2 4854	. . . . . . . 8 ⊢ ({𝐷, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶)
21	18, 20	biimtrdi 253	. . . . . . 7 ⊢ (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶))
22	21	com12 32	. . . . . 6 ⊢ ({𝐴, 𝐵} = {𝐷, 𝐶} → (𝐴 = 𝐷 → 𝐵 = 𝐶))
23	16, 22	sylbi 217	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷 → 𝐵 = 𝐶))
24	23	ancld 550	. . . 4 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷 → (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
25	14, 24	orim12d 966	. . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
26	6, 25	mpd 15	. 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
27		preq12 4740	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
28		preq12 4740	. . . 4 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
29		prcom 4737	. . . 4 ⊢ {𝐷, 𝐶} = {𝐶, 𝐷}
30	28, 29	eqtrdi 2791	. . 3 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
31	27, 30	jaoi 857	. 2 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → {𝐴, 𝐵} = {𝐶, 𝐷})
32	26, 31	impbii 209	1 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {cpr 4633

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634

This theorem is referenced by:  opthpr  4856  preq12bg  4858  opeqpr  5515  opthhausdorff0  5528  axlowdimlem13  28984  upgrwlkdvdelem  29769  altopthsn  35943  sprsymrelfolem2  47418  prproropf1olem4  47431  reuopreuprim  47451  usgrexmpl2trifr  47932  rrx2xpref1o  48568