Theorem prel12 4383

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

Step	Hyp	Ref	Expression
1		preqr1.a	. . . . 5 ⊢ 𝐴 ∈ V
2	1	prid1 4297	. . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		eleq2 2690	. . . 4 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
4	2, 3	mpbii 223	. . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
5		preqr1.b	. . . . 5 ⊢ 𝐵 ∈ V
6	5	prid2 4298	. . . 4 ⊢ 𝐵 ∈ {𝐴, 𝐵}
7		eleq2 2690	. . . 4 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐵 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐶, 𝐷}))
8	6, 7	mpbii 223	. . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 ∈ {𝐶, 𝐷})
9	4, 8	jca 554	. 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))
10	1	elpr 4198	. . . 4 ⊢ (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
11		eqeq2 2633	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐷))
12	11	notbid 308	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐷))
13		orel2 398	. . . . . . . . . . 11 ⊢ (¬ 𝐴 = 𝐷 → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) → 𝐴 = 𝐶))
14	12, 13	syl6bi 243	. . . . . . . . . 10 ⊢ (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) → 𝐴 = 𝐶)))
15	14	impd 447	. . . . . . . . 9 ⊢ (𝐵 = 𝐷 → ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) → 𝐴 = 𝐶))
16	15	com12 32	. . . . . . . 8 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) → (𝐵 = 𝐷 → 𝐴 = 𝐶))
17	16	ancrd 577	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) → (𝐵 = 𝐷 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
18		eqeq2 2633	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐶 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶))
19	18	notbid 308	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐶))
20		orel1 397	. . . . . . . . . . 11 ⊢ (¬ 𝐴 = 𝐶 → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) → 𝐴 = 𝐷))
21	19, 20	syl6bi 243	. . . . . . . . . 10 ⊢ (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) → 𝐴 = 𝐷)))
22	21	impd 447	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) → 𝐴 = 𝐷))
23	22	com12 32	. . . . . . . 8 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) → (𝐵 = 𝐶 → 𝐴 = 𝐷))
24	23	ancrd 577	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) → (𝐵 = 𝐶 → (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
25	17, 24	orim12d 883	. . . . . 6 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) → ((𝐵 = 𝐷 ∨ 𝐵 = 𝐶) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
26	5	elpr 4198	. . . . . . 7 ⊢ (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))
27		orcom 402	. . . . . . 7 ⊢ ((𝐵 = 𝐶 ∨ 𝐵 = 𝐷) ↔ (𝐵 = 𝐷 ∨ 𝐵 = 𝐶))
28	26, 27	bitri 264	. . . . . 6 ⊢ (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐷 ∨ 𝐵 = 𝐶))
29		preq12b.c	. . . . . . 7 ⊢ 𝐶 ∈ V
30		preq12b.d	. . . . . . 7 ⊢ 𝐷 ∈ V
31	1, 5, 29, 30	preq12b 4382	. . . . . 6 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
32	25, 28, 31	3imtr4g 285	. . . . 5 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
33	32	ex 450	. . . 4 ⊢ (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
34	10, 33	syl5bi 232	. . 3 ⊢ (¬ 𝐴 = 𝐵 → (𝐴 ∈ {𝐶, 𝐷} → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
35	34	impd 447	. 2 ⊢ (¬ 𝐴 = 𝐵 → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) → {𝐴, 𝐵} = {𝐶, 𝐷}))
36	9, 35	impbid2 216	1 ⊢ (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  prel12g  4387  dfac2  8953