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Theorem erth 7950
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 474 . . . . . . 7 ((𝜑𝐴𝑅𝐵) → 𝜑)
2 erth.1 . . . . . . . . 9 (𝜑𝑅 Er 𝑋)
32ersymb 7917 . . . . . . . 8 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
43biimpa 502 . . . . . . 7 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51, 4jca 555 . . . . . 6 ((𝜑𝐴𝑅𝐵) → (𝜑𝐵𝑅𝐴))
62ertr 7918 . . . . . . 7 (𝜑 → ((𝐵𝑅𝐴𝐴𝑅𝑥) → 𝐵𝑅𝑥))
76impl 651 . . . . . 6 (((𝜑𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
85, 7sylan 489 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
92ertr 7918 . . . . . 6 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝑥) → 𝐴𝑅𝑥))
109impl 651 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
118, 10impbida 913 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝐴𝑅𝑥𝐵𝑅𝑥))
12 vex 3335 . . . . 5 𝑥 ∈ V
13 erth.2 . . . . . 6 (𝜑𝐴𝑋)
1413adantr 472 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐴𝑋)
15 elecg 7944 . . . . 5 ((𝑥 ∈ V ∧ 𝐴𝑋) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
1612, 14, 15sylancr 698 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
17 errel 7912 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
182, 17syl 17 . . . . . 6 (𝜑 → Rel 𝑅)
19 brrelex2 5306 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
2018, 19sylan 489 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐵 ∈ V)
21 elecg 7944 . . . . 5 ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2212, 20, 21sylancr 698 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2311, 16, 223bitr4d 300 . . 3 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑅))
2423eqrdv 2750 . 2 ((𝜑𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
252adantr 472 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝑅 Er 𝑋)
262, 13erref 7923 . . . . . . 7 (𝜑𝐴𝑅𝐴)
2726adantr 472 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
2813adantr 472 . . . . . . 7 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑋)
29 elecg 7944 . . . . . . 7 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
3028, 28, 29syl2anc 696 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
3127, 30mpbird 247 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
32 simpr 479 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
3331, 32eleqtrd 2833 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
3425, 32ereldm 7949 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴𝑋𝐵𝑋))
3528, 34mpbid 222 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑋)
36 elecg 7944 . . . . 5 ((𝐴𝑋𝐵𝑋) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3728, 35, 36syl2anc 696 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3833, 37mpbid 222 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
3925, 38ersym 7915 . 2 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
4024, 39impbida 913 1 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  Vcvv 3332   class class class wbr 4796  Rel wrel 5263   Er wer 7900  [cec 7901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-er 7903  df-ec 7905
This theorem is referenced by:  erth2  7951  erthi  7952  qliftfun  7991  eroveu  8001  eceqoveq  8011  enreceq  10071  prsrlem1  10077  ercpbllem  16402  orbsta  17938  sylow2blem3  18229  frgpnabllem2  18469  zndvds  20092  qustgpopn  22116  qustgphaus  22119  pi1xfrf  23045  pi1cof  23051  pstmxmet  30241  sconnpi1  31520  topfneec2  32649
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