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

Theorem erth 6336
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 (𝜑𝐴𝑋)
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 107 . . . . . . 7 ((𝜑𝐴𝑅𝐵) → 𝜑)
2 erth.1 . . . . . . . . 9 (𝜑𝑅 Er 𝑋)
32ersymb 6306 . . . . . . . 8 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
43biimpa 290 . . . . . . 7 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51, 4jca 300 . . . . . 6 ((𝜑𝐴𝑅𝐵) → (𝜑𝐵𝑅𝐴))
62ertr 6307 . . . . . . 7 (𝜑 → ((𝐵𝑅𝐴𝐴𝑅𝑥) → 𝐵𝑅𝑥))
76impl 372 . . . . . 6 (((𝜑𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
85, 7sylan 277 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
92ertr 6307 . . . . . 6 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝑥) → 𝐴𝑅𝑥))
109impl 372 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
118, 10impbida 563 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝐴𝑅𝑥𝐵𝑅𝑥))
12 vex 2622 . . . . 5 𝑥 ∈ V
13 erth.2 . . . . . 6 (𝜑𝐴𝑋)
1413adantr 270 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐴𝑋)
15 elecg 6330 . . . . 5 ((𝑥 ∈ V ∧ 𝐴𝑋) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
1612, 14, 15sylancr 405 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
17 errel 6301 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
182, 17syl 14 . . . . . 6 (𝜑 → Rel 𝑅)
19 brrelex2 4478 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
2018, 19sylan 277 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐵 ∈ V)
21 elecg 6330 . . . . 5 ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2212, 20, 21sylancr 405 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2311, 16, 223bitr4d 218 . . 3 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑅))
2423eqrdv 2086 . 2 ((𝜑𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
252adantr 270 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝑅 Er 𝑋)
262, 13erref 6312 . . . . . . 7 (𝜑𝐴𝑅𝐴)
2726adantr 270 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
2813adantr 270 . . . . . . 7 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑋)
29 elecg 6330 . . . . . . 7 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
3028, 28, 29syl2anc 403 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
3127, 30mpbird 165 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
32 simpr 108 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
3331, 32eleqtrd 2166 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
3425, 32ereldm 6335 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴𝑋𝐵𝑋))
3528, 34mpbid 145 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑋)
36 elecg 6330 . . . . 5 ((𝐴𝑋𝐵𝑋) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3728, 35, 36syl2anc 403 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3833, 37mpbid 145 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
3925, 38ersym 6304 . 2 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
4024, 39impbida 563 1 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  Vcvv 2619   class class class wbr 3845  Rel wrel 4443   Er wer 6289  [cec 6290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-er 6292  df-ec 6294
This theorem is referenced by:  erth2  6337  erthi  6338  qliftfun  6374  eroveu  6383  th3qlem1  6394  enqeceq  6918  enq0eceq  6996  nnnq0lem1  7005  enreceq  7282  prsrlem1  7288
  Copyright terms: Public domain W3C validator