Theorem ssprsseq 3840

Description: A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.)

Assertion

Ref	Expression
ssprsseq	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))

Proof of Theorem ssprsseq

Step	Hyp	Ref	Expression
1		ssprss 3839	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))
2	1	3adant3 1044	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))
3		eqneqall 2413	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
4		eqtr3 2251	. . . . . . . 8 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
5	3, 4	syl11 31	. . . . . . 7 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
6	5	3ad2ant3 1047	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
7	6	com12 30	. . . . 5 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
8		preq12 3754	. . . . . . 7 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
9		prcom 3751	. . . . . . 7 ⊢ {𝐷, 𝐶} = {𝐶, 𝐷}
10	8, 9	eqtrdi 2280	. . . . . 6 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
11	10	a1d 22	. . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
12		preq12 3754	. . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
13	12	a1d 22	. . . . 5 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
14		eqtr3 2251	. . . . . . . 8 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → 𝐴 = 𝐵)
15	3, 14	syl11 31	. . . . . . 7 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
16	15	3ad2ant3 1047	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
17	16	com12 30	. . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
18	7, 11, 13, 17	ccase 973	. . . 4 ⊢ (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
19	18	com12 30	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
20	2, 19	sylbid 150	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
21		eqimss 3282	. 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → {𝐴, 𝐵} ⊆ {𝐶, 𝐷})
22	20, 21	impbid1 142	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202   ≠ wne 2403   ⊆ wss 3201  {cpr 3674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680

This theorem is referenced by:  upgredgpr  16073