MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  preq12b Structured version   Visualization version   GIF version

Theorem preq12b 4413
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 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))

Proof of Theorem preq12b
StepHypRef Expression
1 preqr1.a . . . . . 6 𝐴 ∈ V
21prid1 4329 . . . . 5 𝐴 ∈ {𝐴, 𝐵}
3 eleq2 2719 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
42, 3mpbii 223 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
51elpr 4231 . . . 4 (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷))
64, 5sylib 208 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐴 = 𝐷))
7 preq1 4300 . . . . . . . 8 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
87eqeq1d 2653 . . . . . . 7 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
9 preqr1.b . . . . . . . 8 𝐵 ∈ V
10 preq12b.d . . . . . . . 8 𝐷 ∈ V
119, 10preqr2 4412 . . . . . . 7 ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
128, 11syl6bi 243 . . . . . 6 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
1312com12 32 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷))
1413ancld 575 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷)))
15 prcom 4299 . . . . . . 7 {𝐶, 𝐷} = {𝐷, 𝐶}
1615eqeq2i 2663 . . . . . 6 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
17 preq1 4300 . . . . . . . . 9 (𝐴 = 𝐷 → {𝐴, 𝐵} = {𝐷, 𝐵})
1817eqeq1d 2653 . . . . . . . 8 (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} ↔ {𝐷, 𝐵} = {𝐷, 𝐶}))
19 preq12b.c . . . . . . . . 9 𝐶 ∈ V
209, 19preqr2 4412 . . . . . . . 8 ({𝐷, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶)
2118, 20syl6bi 243 . . . . . . 7 (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶))
2221com12 32 . . . . . 6 ({𝐴, 𝐵} = {𝐷, 𝐶} → (𝐴 = 𝐷𝐵 = 𝐶))
2316, 22sylbi 207 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷𝐵 = 𝐶))
2423ancld 575 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷 → (𝐴 = 𝐷𝐵 = 𝐶)))
2514, 24orim12d 901 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐴 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
266, 25mpd 15 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
27 preq12 4302 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
28 preq12 4302 . . . 4 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
29 prcom 4299 . . . 4 {𝐷, 𝐶} = {𝐶, 𝐷}
3028, 29syl6eq 2701 . . 3 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
3127, 30jaoi 393 . 2 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → {𝐴, 𝐵} = {𝐶, 𝐷})
3226, 31impbii 199 1 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213
This theorem is referenced by:  prel12  4414  opthpr  4415  preq12bg  4417  preqsn  4424  preqsnOLD  4425  opeqpr  4997  preleq  8552  axlowdimlem13  25879  upgrwlkdvdelem  26688  altopthsn  32193  sprsymrelfolem2  42068
  Copyright terms: Public domain W3C validator