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Theorem prel12 3589
Description: Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
Hypotheses
Ref Expression
preq12b.1 𝐴 ∈ V
preq12b.2 𝐵 ∈ V
preq12b.3 𝐶 ∈ V
preq12b.4 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 preq12b.1 . . . . 5 𝐴 ∈ V
21prid1 3522 . . . 4 𝐴 ∈ {𝐴, 𝐵}
3 eleq2 2146 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
42, 3mpbii 146 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
5 preq12b.2 . . . . 5 𝐵 ∈ V
65prid2 3523 . . . 4 𝐵 ∈ {𝐴, 𝐵}
7 eleq2 2146 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐵 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐶, 𝐷}))
86, 7mpbii 146 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 ∈ {𝐶, 𝐷})
94, 8jca 300 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))
101elpr 3443 . . . 4 (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷))
11 eqeq2 2092 . . . . . . . . . . . 12 (𝐵 = 𝐷 → (𝐴 = 𝐵𝐴 = 𝐷))
1211notbid 625 . . . . . . . . . . 11 (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐷))
13 orel2 678 . . . . . . . . . . 11 𝐴 = 𝐷 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐶))
1412, 13syl6bi 161 . . . . . . . . . 10 (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐶)))
1514com3l 80 . . . . . . . . 9 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 = 𝐷𝐴 = 𝐶)))
1615imp 122 . . . . . . . 8 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐷𝐴 = 𝐶))
1716ancrd 319 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐷 → (𝐴 = 𝐶𝐵 = 𝐷)))
18 eqeq2 2092 . . . . . . . . . . . 12 (𝐵 = 𝐶 → (𝐴 = 𝐵𝐴 = 𝐶))
1918notbid 625 . . . . . . . . . . 11 (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐶))
20 orel1 677 . . . . . . . . . . 11 𝐴 = 𝐶 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐷))
2119, 20syl6bi 161 . . . . . . . . . 10 (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐷)))
2221com3l 80 . . . . . . . . 9 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 = 𝐶𝐴 = 𝐷)))
2322imp 122 . . . . . . . 8 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐶𝐴 = 𝐷))
2423ancrd 319 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐶 → (𝐴 = 𝐷𝐵 = 𝐶)))
2517, 24orim12d 733 . . . . . 6 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → ((𝐵 = 𝐷𝐵 = 𝐶) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
265elpr 3443 . . . . . . 7 (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶𝐵 = 𝐷))
27 orcom 680 . . . . . . 7 ((𝐵 = 𝐶𝐵 = 𝐷) ↔ (𝐵 = 𝐷𝐵 = 𝐶))
2826, 27bitri 182 . . . . . 6 (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐷𝐵 = 𝐶))
29 preq12b.3 . . . . . . 7 𝐶 ∈ V
30 preq12b.4 . . . . . . 7 𝐷 ∈ V
311, 5, 29, 30preq12b 3588 . . . . . 6 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
3225, 28, 313imtr4g 203 . . . . 5 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
3332ex 113 . . . 4 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
3410, 33syl5bi 150 . . 3 𝐴 = 𝐵 → (𝐴 ∈ {𝐶, 𝐷} → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
3534impd 251 . 2 𝐴 = 𝐵 → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) → {𝐴, 𝐵} = {𝐶, 𝐷}))
369, 35impbid2 141 1 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662   = wceq 1285  wcel 1434  Vcvv 2612  {cpr 3423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429
This theorem is referenced by: (None)
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