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Theorem ssprsseq 4764
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}))

Proof of Theorem ssprsseq
StepHypRef Expression
1 ssprss 4763 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
213adant3 1131 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
3 eqneqall 2956 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
4 eqtr3 2766 . . . . . . . 8 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
53, 4syl11 33 . . . . . . 7 (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
653ad2ant3 1134 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
76com12 32 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐶) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
8 preq12 4677 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
9 prcom 4674 . . . . . . 7 {𝐷, 𝐶} = {𝐶, 𝐷}
108, 9eqtrdi 2796 . . . . . 6 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
1110a1d 25 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
12 preq12 4677 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
1312a1d 25 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
14 eqtr3 2766 . . . . . . . 8 ((𝐴 = 𝐷𝐵 = 𝐷) → 𝐴 = 𝐵)
153, 14syl11 33 . . . . . . 7 (𝐴𝐵 → ((𝐴 = 𝐷𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
16153ad2ant3 1134 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐷𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
1716com12 32 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐷) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
187, 11, 13, 17ccase 1035 . . . 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 239 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
21 eqimss 3982 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → {𝐴, 𝐵} ⊆ {𝐶, 𝐷})
2220, 21impbid1 224 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844  w3a 1086   = wceq 1542  wcel 2110  wne 2945  wss 3892  {cpr 4569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-v 3433  df-un 3897  df-in 3899  df-ss 3909  df-sn 4568  df-pr 4570
This theorem is referenced by:  upgredgpr  27510  cyc3genpmlem  31414
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