Theorem ssprsseq 4760

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 4759	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))
2	1	3adant3 1128	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))
3		eqneqall 3029	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
4		eqtr3 2845	. . . . . . . 8 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
5	3, 4	syl11 33	. . . . . . 7 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
6	5	3ad2ant3 1131	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
7	6	com12 32	. . . . 5 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
8		preq12 4673	. . . . . . 7 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
9		prcom 4670	. . . . . . 7 ⊢ {𝐷, 𝐶} = {𝐶, 𝐷}
10	8, 9	syl6eq 2874	. . . . . 6 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
11	10	a1d 25	. . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
12		preq12 4673	. . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
13	12	a1d 25	. . . . 5 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
14		eqtr3 2845	. . . . . . . 8 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → 𝐴 = 𝐵)
15	3, 14	syl11 33	. . . . . . 7 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
16	15	3ad2ant3 1131	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
17	16	com12 32	. . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
18	7, 11, 13, 17	ccase 1032	. . . 4 ⊢ (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
19	18	com12 32	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
20	2, 19	sylbid 242	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
21		eqimss 4025	. 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → {𝐴, 𝐵} ⊆ {𝐶, 𝐷})
22	20, 21	impbid1 227	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   ⊆ wss 3938  {cpr 4571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-v 3498  df-un 3943  df-in 3945  df-ss 3954  df-sn 4570  df-pr 4572

This theorem is referenced by:  upgredgpr  26929  cyc3genpmlem  30795