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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 4331 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
213adant3 1079 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
3 eqneqall 2801 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
4 eqtr3 2642 . . . . . . . 8 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
53, 4syl11 33 . . . . . . 7 (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
653ad2ant3 1082 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
76com12 32 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐶) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
8 preq12 4247 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
9 prcom 4244 . . . . . . 7 {𝐷, 𝐶} = {𝐶, 𝐷}
108, 9syl6eq 2671 . . . . . 6 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
1110a1d 25 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
12 preq12 4247 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
1312a1d 25 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
14 eqtr3 2642 . . . . . . . 8 ((𝐴 = 𝐷𝐵 = 𝐷) → 𝐴 = 𝐵)
153, 14syl11 33 . . . . . . 7 (𝐴𝐵 → ((𝐴 = 𝐷𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
16153ad2ant3 1082 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐷𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
1716com12 32 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐷) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
187, 11, 13, 17ccase 986 . . . 4 (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
1918com12 32 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷}))
202, 19sylbid 230 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
21 eqimss 3642 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → {𝐴, 𝐵} ⊆ {𝐶, 𝐷})
2220, 21impbid1 215 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790   ⊆ wss 3560  {cpr 4157 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3192  df-un 3565  df-in 3567  df-ss 3574  df-sn 4156  df-pr 4158 This theorem is referenced by:  upgredgpr  25966
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