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Theorem preleqALT 9058
Description: Alternate proof of preleq 9057, not based on preleqg 9056: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
preleq.b 𝐵 ∈ V
preleqALT.d 𝐷 ∈ V
Assertion
Ref Expression
preleqALT (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem preleqALT
StepHypRef Expression
1 preleq.b . . . . . . . . . 10 𝐵 ∈ V
21jctr 527 . . . . . . . . 9 (𝐴𝐵 → (𝐴𝐵𝐵 ∈ V))
3 preleqALT.d . . . . . . . . . 10 𝐷 ∈ V
43jctr 527 . . . . . . . . 9 (𝐶𝐷 → (𝐶𝐷𝐷 ∈ V))
5 preq12bg 4760 . . . . . . . . 9 (((𝐴𝐵𝐵 ∈ V) ∧ (𝐶𝐷𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
62, 4, 5syl2an 597 . . . . . . . 8 ((𝐴𝐵𝐶𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
76biimpa 479 . . . . . . 7 (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
87ord 860 . . . . . 6 (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (¬ (𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 = 𝐷𝐵 = 𝐶)))
9 en2lp 9047 . . . . . . 7 ¬ (𝐷𝐶𝐶𝐷)
10 eleq12 2900 . . . . . . . 8 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴𝐵𝐷𝐶))
1110anbi1d 631 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴𝐵𝐶𝐷) ↔ (𝐷𝐶𝐶𝐷)))
129, 11mtbiri 329 . . . . . 6 ((𝐴 = 𝐷𝐵 = 𝐶) → ¬ (𝐴𝐵𝐶𝐷))
138, 12syl6 35 . . . . 5 (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (¬ (𝐴 = 𝐶𝐵 = 𝐷) → ¬ (𝐴𝐵𝐶𝐷)))
1413con4d 115 . . . 4 (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → ((𝐴𝐵𝐶𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
1514ex 415 . . 3 ((𝐴𝐵𝐶𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴𝐵𝐶𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))))
1615pm2.43a 54 . 2 ((𝐴𝐵𝐶𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷)))
1716imp 409 1 (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1537  wcel 2114  Vcvv 3473  {cpr 4545
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 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-reg 9034
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-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-eprel 5441  df-fr 5490
This theorem is referenced by: (None)
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