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Theorem erth 6086
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 6056 . . . . . . . 8 (φ → (A𝑅BB𝑅A))
43biimpa 280 . . . . . . 7 ((φ A𝑅B) → B𝑅A)
51, 4jca 290 . . . . . 6 ((φ A𝑅B) → (φ B𝑅A))
62ertr 6057 . . . . . . 7 (φ → ((B𝑅A A𝑅x) → B𝑅x))
76impl 362 . . . . . 6 (((φ B𝑅A) A𝑅x) → B𝑅x)
85, 7sylan 267 . . . . 5 (((φ A𝑅B) A𝑅x) → B𝑅x)
92ertr 6057 . . . . . 6 (φ → ((A𝑅B B𝑅x) → A𝑅x))
109impl 362 . . . . 5 (((φ A𝑅B) B𝑅x) → A𝑅x)
118, 10impbida 528 . . . 4 ((φ A𝑅B) → (A𝑅xB𝑅x))
12 vex 2554 . . . . 5 x V
13 erth.2 . . . . . 6 (φA 𝑋)
1413adantr 261 . . . . 5 ((φ A𝑅B) → A 𝑋)
15 elecg 6080 . . . . 5 ((x V A 𝑋) → (x [A]𝑅A𝑅x))
1612, 14, 15sylancr 393 . . . 4 ((φ A𝑅B) → (x [A]𝑅A𝑅x))
17 errel 6051 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
182, 17syl 14 . . . . . 6 (φ → Rel 𝑅)
19 brrelex2 4326 . . . . . 6 ((Rel 𝑅 A𝑅B) → B V)
2018, 19sylan 267 . . . . 5 ((φ A𝑅B) → B V)
21 elecg 6080 . . . . 5 ((x V B V) → (x [B]𝑅B𝑅x))
2212, 20, 21sylancr 393 . . . 4 ((φ A𝑅B) → (x [B]𝑅B𝑅x))
2311, 16, 223bitr4d 209 . . 3 ((φ A𝑅B) → (x [A]𝑅x [B]𝑅))
2423eqrdv 2035 . 2 ((φ A𝑅B) → [A]𝑅 = [B]𝑅)
252adantr 261 . . 3 ((φ [A]𝑅 = [B]𝑅) → 𝑅 Er 𝑋)
262, 13erref 6062 . . . . . . 7 (φA𝑅A)
2726adantr 261 . . . . . 6 ((φ [A]𝑅 = [B]𝑅) → A𝑅A)
2813adantr 261 . . . . . . 7 ((φ [A]𝑅 = [B]𝑅) → A 𝑋)
29 elecg 6080 . . . . . . 7 ((A 𝑋 A 𝑋) → (A [A]𝑅A𝑅A))
3028, 28, 29syl2anc 391 . . . . . 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 2113 . . . 4 ((φ [A]𝑅 = [B]𝑅) → A [B]𝑅)
3425, 32ereldm 6085 . . . . . 6 ((φ [A]𝑅 = [B]𝑅) → (A 𝑋B 𝑋))
3528, 34mpbid 135 . . . . 5 ((φ [A]𝑅 = [B]𝑅) → B 𝑋)
36 elecg 6080 . . . . 5 ((A 𝑋 B 𝑋) → (A [B]𝑅B𝑅A))
3728, 35, 36syl2anc 391 . . . 4 ((φ [A]𝑅 = [B]𝑅) → (A [B]𝑅B𝑅A))
3833, 37mpbid 135 . . 3 ((φ [A]𝑅 = [B]𝑅) → B𝑅A)
3925, 38ersym 6054 . 2 ((φ [A]𝑅 = [B]𝑅) → A𝑅B)
4024, 39impbida 528 1 (φ → (A𝑅B ↔ [A]𝑅 = [B]𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551   class class class wbr 3755  Rel wrel 4293   Er wer 6039  [cec 6040
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 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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-er 6042  df-ec 6044
This theorem is referenced by:  erth2  6087  erthi  6088  qliftfun  6124  eroveu  6133  th3qlem1  6144  enqeceq  6343  enq0eceq  6419  nnnq0lem1  6428  enreceq  6624  prsrlem1  6630
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