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Theorem preleq 4233
Description: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
Hypotheses
Ref Expression
preleq.1 A V
preleq.2 B V
preleq.3 𝐶 V
preleq.4 𝐷 V
Assertion
Ref Expression
preleq (((A B 𝐶 𝐷) {A, B} = {𝐶, 𝐷}) → (A = 𝐶 B = 𝐷))

Proof of Theorem preleq
StepHypRef Expression
1 en2lp 4232 . . . . 5 ¬ (𝐷 𝐶 𝐶 𝐷)
2 eleq12 2099 . . . . . 6 ((A = 𝐷 B = 𝐶) → (A B𝐷 𝐶))
32anbi1d 438 . . . . 5 ((A = 𝐷 B = 𝐶) → ((A B 𝐶 𝐷) ↔ (𝐷 𝐶 𝐶 𝐷)))
41, 3mtbiri 599 . . . 4 ((A = 𝐷 B = 𝐶) → ¬ (A B 𝐶 𝐷))
54con2i 557 . . 3 ((A B 𝐶 𝐷) → ¬ (A = 𝐷 B = 𝐶))
65adantr 261 . 2 (((A B 𝐶 𝐷) {A, B} = {𝐶, 𝐷}) → ¬ (A = 𝐷 B = 𝐶))
7 preleq.1 . . . . 5 A V
8 preleq.2 . . . . 5 B V
9 preleq.3 . . . . 5 𝐶 V
10 preleq.4 . . . . 5 𝐷 V
117, 8, 9, 10preq12b 3532 . . . 4 ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)))
1211biimpi 113 . . 3 ({A, B} = {𝐶, 𝐷} → ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)))
1312adantl 262 . 2 (((A B 𝐶 𝐷) {A, B} = {𝐶, 𝐷}) → ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)))
146, 13ecased 1238 1 (((A B 𝐶 𝐷) {A, B} = {𝐶, 𝐷}) → (A = 𝐶 B = 𝐷))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 628   = wceq 1242   wcel 1390  Vcvv 2551  {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  opthreg  4234
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