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Theorem erth 6061
Description: Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
Hypotheses
Ref Expression
erth.1 (φ𝑅 Er 𝑋)
erth.2 (φA 𝑋)
Assertion
Ref Expression
erth (φ → (A𝑅B ↔ [A]𝑅 = [B]𝑅))

Proof of Theorem erth
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simpl 102 . . . . . . 7 ((φ A𝑅B) → φ)
2 erth.1 . . . . . . . . 9 (φ𝑅 Er 𝑋)
32ersymb 6031 . . . . . . . 8 (φ → (A𝑅BB𝑅A))
43biimpa 280 . . . . . . 7 ((φ A𝑅B) → B𝑅A)
51, 4jca 290 . . . . . 6 ((φ A𝑅B) → (φ B𝑅A))
62ertr 6032 . . . . . . 7 (φ → ((B𝑅A A𝑅x) → B𝑅x))
76impl 362 . . . . . 6 (((φ B𝑅A) A𝑅x) → B𝑅x)
85, 7sylan 267 . . . . 5 (((φ A𝑅B) A𝑅x) → B𝑅x)
92ertr 6032 . . . . . 6 (φ → ((A𝑅B B𝑅x) → A𝑅x))
109impl 362 . . . . 5 (((φ A𝑅B) B𝑅x) → A𝑅x)
118, 10impbida 515 . . . 4 ((φ A𝑅B) → (A𝑅xB𝑅x))
12 vex 2538 . . . . 5 x V
13 erth.2 . . . . . 6 (φA 𝑋)
1413adantr 261 . . . . 5 ((φ A𝑅B) → A 𝑋)
15 elecg 6055 . . . . 5 ((x V A 𝑋) → (x [A]𝑅A𝑅x))
1612, 14, 15sylancr 395 . . . 4 ((φ A𝑅B) → (x [A]𝑅A𝑅x))
17 errel 6026 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
182, 17syl 14 . . . . . 6 (φ → Rel 𝑅)
19 brrelex2 4310 . . . . . 6 ((Rel 𝑅 A𝑅B) → B V)
2018, 19sylan 267 . . . . 5 ((φ A𝑅B) → B V)
21 elecg 6055 . . . . 5 ((x V B V) → (x [B]𝑅B𝑅x))
2212, 20, 21sylancr 395 . . . 4 ((φ A𝑅B) → (x [B]𝑅B𝑅x))
2311, 16, 223bitr4d 209 . . 3 ((φ A𝑅B) → (x [A]𝑅x [B]𝑅))
2423eqrdv 2020 . 2 ((φ A𝑅B) → [A]𝑅 = [B]𝑅)
252adantr 261 . . 3 ((φ [A]𝑅 = [B]𝑅) → 𝑅 Er 𝑋)
262, 13erref 6037 . . . . . . 7 (φA𝑅A)
2726adantr 261 . . . . . 6 ((φ [A]𝑅 = [B]𝑅) → A𝑅A)
2813adantr 261 . . . . . . 7 ((φ [A]𝑅 = [B]𝑅) → A 𝑋)
29 elecg 6055 . . . . . . 7 ((A 𝑋 A 𝑋) → (A [A]𝑅A𝑅A))
3028, 28, 29syl2anc 393 . . . . . 6 ((φ [A]𝑅 = [B]𝑅) → (A [A]𝑅A𝑅A))
3127, 30mpbird 156 . . . . 5 ((φ [A]𝑅 = [B]𝑅) → A [A]𝑅)
32 simpr 103 . . . . 5 ((φ [A]𝑅 = [B]𝑅) → [A]𝑅 = [B]𝑅)
3331, 32eleqtrd 2098 . . . 4 ((φ [A]𝑅 = [B]𝑅) → A [B]𝑅)
3425, 32ereldm 6060 . . . . . 6 ((φ [A]𝑅 = [B]𝑅) → (A 𝑋B 𝑋))
3528, 34mpbid 135 . . . . 5 ((φ [A]𝑅 = [B]𝑅) → B 𝑋)
36 elecg 6055 . . . . 5 ((A 𝑋 B 𝑋) → (A [B]𝑅B𝑅A))
3728, 35, 36syl2anc 393 . . . 4 ((φ [A]𝑅 = [B]𝑅) → (A [B]𝑅B𝑅A))
3833, 37mpbid 135 . . 3 ((φ [A]𝑅 = [B]𝑅) → B𝑅A)
3925, 38ersym 6029 . 2 ((φ [A]𝑅 = [B]𝑅) → A𝑅B)
4024, 39impbida 515 1 (φ → (A𝑅B ↔ [A]𝑅 = [B]𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535   class class class wbr 3738  Rel wrel 4277   Er wer 6014  [cec 6015
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-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-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-er 6017  df-ec 6019
This theorem is referenced by:  erth2  6062  erthi  6063  qliftfun  6099  eroveu  6108  th3qlem1  6119  enqeceq  6218  enq0eceq  6292  nnnq0lem1  6301  enreceq  6483  prsrlem1  6489
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