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Theorem erth 8331
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 erth.1 . . . . . . . 8 (𝜑𝑅 Er 𝑋)
21ersymb 8296 . . . . . . 7 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
32biimpa 477 . . . . . 6 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
41ertr 8297 . . . . . . 7 (𝜑 → ((𝐵𝑅𝐴𝐴𝑅𝑥) → 𝐵𝑅𝑥))
54impl 456 . . . . . 6 (((𝜑𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
63, 5syldanl 601 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
71ertr 8297 . . . . . 6 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝑥) → 𝐴𝑅𝑥))
87impl 456 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
96, 8impbida 797 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝐴𝑅𝑥𝐵𝑅𝑥))
10 vex 3502 . . . . 5 𝑥 ∈ V
11 erth.2 . . . . . 6 (𝜑𝐴𝑋)
1211adantr 481 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐴𝑋)
13 elecg 8325 . . . . 5 ((𝑥 ∈ V ∧ 𝐴𝑋) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
1410, 12, 13sylancr 587 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
15 errel 8291 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
161, 15syl 17 . . . . . 6 (𝜑 → Rel 𝑅)
17 brrelex2 5604 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
1816, 17sylan 580 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐵 ∈ V)
19 elecg 8325 . . . . 5 ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2010, 18, 19sylancr 587 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
219, 14, 203bitr4d 312 . . 3 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑅))
2221eqrdv 2823 . 2 ((𝜑𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
231adantr 481 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝑅 Er 𝑋)
241, 11erref 8302 . . . . . . 7 (𝜑𝐴𝑅𝐴)
2524adantr 481 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
2611adantr 481 . . . . . . 7 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑋)
27 elecg 8325 . . . . . . 7 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
2826, 26, 27syl2anc 584 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
2925, 28mpbird 258 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
30 simpr 485 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
3129, 30eleqtrd 2919 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
3223, 30ereldm 8330 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴𝑋𝐵𝑋))
3326, 32mpbid 233 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑋)
34 elecg 8325 . . . . 5 ((𝐴𝑋𝐵𝑋) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3526, 33, 34syl2anc 584 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3631, 35mpbid 233 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
3723, 36ersym 8294 . 2 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
3822, 37impbida 797 1 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  Vcvv 3499   class class class wbr 5062  Rel wrel 5558   Er wer 8279  [cec 8280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-er 8282  df-ec 8284
This theorem is referenced by:  erth2  8332  erthi  8333  qliftfun  8375  eroveu  8385  eceqoveq  8395  enreceq  10480  prsrlem1  10486  ercpbllem  16813  orbsta  18375  sylow2blem3  18669  frgpnabllem2  18916  zndvds  20612  qustgpopn  22643  qustgphaus  22646  pi1xfrf  23572  pi1cof  23578  tgjustr  26174  qusvscpbl  30834  eqg0el  30840  pstmxmet  31023  sconnpi1  32370  topfneec2  33588
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