ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  preleq Structured version   GIF version

Theorem preleq 4217
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 4216 . . . . 5 ¬ (𝐷 𝐶 𝐶 𝐷)
2 eleq12 2084 . . . . . 6 ((A = 𝐷 B = 𝐶) → (A B𝐷 𝐶))
32anbi1d 441 . . . . 5 ((A = 𝐷 B = 𝐶) → ((A B 𝐶 𝐷) ↔ (𝐷 𝐶 𝐶 𝐷)))
41, 3mtbiri 587 . . . 4 ((A = 𝐷 B = 𝐶) → ¬ (A B 𝐶 𝐷))
54con2i 545 . . 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 3515 . . . 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 1224 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 616   = wceq 1228   wcel 1374  Vcvv 2535  {cpr 3351
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-dif 2897  df-un 2899  df-sn 3356  df-pr 3357
This theorem is referenced by:  opthreg  4218
  Copyright terms: Public domain W3C validator