Theorem opthpr 3534

Description: A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)

Hypotheses

Ref	Expression
preq12b.1	⊢ A ∈ V
preq12b.2	⊢ B ∈ V
preq12b.3	⊢ 𝐶 ∈ V
preq12b.4	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
opthpr	⊢ (A ≠ 𝐷 → ({A, B} = {𝐶, 𝐷} ↔ (A = 𝐶 ∧ B = 𝐷)))

Proof of Theorem opthpr

Step	Hyp	Ref	Expression
1		preq12b.1	. . 3 ⊢ A ∈ V
2		preq12b.2	. . 3 ⊢ B ∈ V
3		preq12b.3	. . 3 ⊢ 𝐶 ∈ V
4		preq12b.4	. . 3 ⊢ 𝐷 ∈ V
5	1, 2, 3, 4	preq12b 3532	. 2 ⊢ ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)))
6		idd 21	. . . 4 ⊢ (A ≠ 𝐷 → ((A = 𝐶 ∧ B = 𝐷) → (A = 𝐶 ∧ B = 𝐷)))
7		df-ne 2203	. . . . . 6 ⊢ (A ≠ 𝐷 ↔ ¬ A = 𝐷)
8		pm2.21 547	. . . . . 6 ⊢ (¬ A = 𝐷 → (A = 𝐷 → (B = 𝐶 → (A = 𝐶 ∧ B = 𝐷))))
9	7, 8	sylbi 114	. . . . 5 ⊢ (A ≠ 𝐷 → (A = 𝐷 → (B = 𝐶 → (A = 𝐶 ∧ B = 𝐷))))
10	9	impd 242	. . . 4 ⊢ (A ≠ 𝐷 → ((A = 𝐷 ∧ B = 𝐶) → (A = 𝐶 ∧ B = 𝐷)))
11	6, 10	jaod 636	. . 3 ⊢ (A ≠ 𝐷 → (((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)) → (A = 𝐶 ∧ B = 𝐷)))
12		orc 632	. . 3 ⊢ ((A = 𝐶 ∧ B = 𝐷) → ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)))
13	11, 12	impbid1 130	. 2 ⊢ (A ≠ 𝐷 → (((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)) ↔ (A = 𝐶 ∧ B = 𝐷)))
14	5, 13	syl5bb 181	1 ⊢ (A ≠ 𝐷 → ({A, B} = {𝐶, 𝐷} ↔ (A = 𝐶 ∧ B = 𝐷)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ≠ wne 2201  Vcvv 2551  {cpr 3368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374

This theorem is referenced by: (None)