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Theorem ssprsseq 4587
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 4586 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
213adant3 1123 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
3 eqneqall 2979 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷}))
4 eqtr3 2800 . . . . . . . 8 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
53, 4syl11 33 . . . . . . 7 (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
653ad2ant3 1126 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}))
76com12 32 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐶) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
8 preq12 4501 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
9 prcom 4498 . . . . . . 7 {𝐷, 𝐶} = {𝐶, 𝐷}
108, 9syl6eq 2829 . . . . . 6 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
1110a1d 25 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
12 preq12 4501 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
1312a1d 25 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
14 eqtr3 2800 . . . . . . . 8 ((𝐴 = 𝐷𝐵 = 𝐷) → 𝐴 = 𝐵)
153, 14syl11 33 . . . . . . 7 (𝐴𝐵 → ((𝐴 = 𝐷𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
16153ad2ant3 1126 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐷𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}))
1716com12 32 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐷) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷}))
187, 11, 13, 17ccase 1021 . . . 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 232 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
21 eqimss 3875 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → {𝐴, 𝐵} ⊆ {𝐶, 𝐷})
2220, 21impbid1 217 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wo 836  w3a 1071   = wceq 1601  wcel 2106  wne 2968  wss 3791  {cpr 4399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-v 3399  df-un 3796  df-in 3798  df-ss 3805  df-sn 4398  df-pr 4400
This theorem is referenced by:  upgredgpr  26491
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