Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  prel12 Structured version   Visualization version   GIF version

Theorem prel12 4351
 Description: Equality of 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
prel12 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))

Proof of Theorem prel12
StepHypRef Expression
1 preqr1.a . . . . 5 𝐴 ∈ V
21prid1 4267 . . . 4 𝐴 ∈ {𝐴, 𝐵}
3 eleq2 2687 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
42, 3mpbii 223 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
5 preqr1.b . . . . 5 𝐵 ∈ V
65prid2 4268 . . . 4 𝐵 ∈ {𝐴, 𝐵}
7 eleq2 2687 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐵 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐶, 𝐷}))
86, 7mpbii 223 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 ∈ {𝐶, 𝐷})
94, 8jca 554 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))
101elpr 4169 . . . 4 (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷))
11 eqeq2 2632 . . . . . . . . . . . 12 (𝐵 = 𝐷 → (𝐴 = 𝐵𝐴 = 𝐷))
1211notbid 308 . . . . . . . . . . 11 (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐷))
13 orel2 398 . . . . . . . . . . 11 𝐴 = 𝐷 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐶))
1412, 13syl6bi 243 . . . . . . . . . 10 (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐶)))
1514impd 447 . . . . . . . . 9 (𝐵 = 𝐷 → ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → 𝐴 = 𝐶))
1615com12 32 . . . . . . . 8 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐷𝐴 = 𝐶))
1716ancrd 576 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐷 → (𝐴 = 𝐶𝐵 = 𝐷)))
18 eqeq2 2632 . . . . . . . . . . . 12 (𝐵 = 𝐶 → (𝐴 = 𝐵𝐴 = 𝐶))
1918notbid 308 . . . . . . . . . . 11 (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐶))
20 orel1 397 . . . . . . . . . . 11 𝐴 = 𝐶 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐷))
2119, 20syl6bi 243 . . . . . . . . . 10 (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐷)))
2221impd 447 . . . . . . . . 9 (𝐵 = 𝐶 → ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → 𝐴 = 𝐷))
2322com12 32 . . . . . . . 8 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐶𝐴 = 𝐷))
2423ancrd 576 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐶 → (𝐴 = 𝐷𝐵 = 𝐶)))
2517, 24orim12d 882 . . . . . 6 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → ((𝐵 = 𝐷𝐵 = 𝐶) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
265elpr 4169 . . . . . . 7 (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶𝐵 = 𝐷))
27 orcom 402 . . . . . . 7 ((𝐵 = 𝐶𝐵 = 𝐷) ↔ (𝐵 = 𝐷𝐵 = 𝐶))
2826, 27bitri 264 . . . . . 6 (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐷𝐵 = 𝐶))
29 preq12b.c . . . . . . 7 𝐶 ∈ V
30 preq12b.d . . . . . . 7 𝐷 ∈ V
311, 5, 29, 30preq12b 4350 . . . . . 6 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
3225, 28, 313imtr4g 285 . . . . 5 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
3332ex 450 . . . 4 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
3410, 33syl5bi 232 . . 3 𝐴 = 𝐵 → (𝐴 ∈ {𝐶, 𝐷} → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
3534impd 447 . 2 𝐴 = 𝐵 → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) → {𝐴, 𝐵} = {𝐶, 𝐷}))
369, 35impbid2 216 1 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186  {cpr 4150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560  df-sn 4149  df-pr 4151 This theorem is referenced by:  prel12g  4355  dfac2  8897
 Copyright terms: Public domain W3C validator