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 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 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))

StepHypRef Expression
1 ssprss 4296 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
213adant3 1074 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
3 eqtr3 2631 . . . . . . . . 9 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
4 eqneqall 2793 . . . . . . . . 9 (𝐴 = 𝐵 → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
53, 4syl 17 . . . . . . . 8 ((𝐴 = 𝐶𝐵 = 𝐶) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
65com12 32 . . . . . . 7 (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
763ad2ant3 1077 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
87com12 32 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐶) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
9 preq12 4214 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
10 prcom 4211 . . . . . . 7 {𝐷, 𝐶} = {𝐶, 𝐷}
119, 10syl6eq 2660 . . . . . 6 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
1211a1d 25 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
13 preq12 4214 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
1413a1d 25 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
15 eqtr3 2631 . . . . . . . . 9 ((𝐴 = 𝐷𝐵 = 𝐷) → 𝐴 = 𝐵)
1615, 4syl 17 . . . . . . . 8 ((𝐴 = 𝐷𝐵 = 𝐷) → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
1716com12 32 . . . . . . 7 (𝐴𝐵 → ((𝐴 = 𝐷𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
18173ad2ant3 1077 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐷𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
1918com12 32 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐷) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
208, 12, 14, 19ccase 984 . . . 4 (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
2120com12 32 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
222, 21sylbid 229 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
23 eqimss 3620 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → {𝐴, 𝐵} ⊆ {𝐶, 𝐷})
2422, 23impbid1 214 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128 This theorem is referenced by:  upgredgpr  25815
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