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Theorem erth 6150
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 102 . . . . . . 7 ((𝜑𝐴𝑅𝐵) → 𝜑)
2 erth.1 . . . . . . . . 9 (𝜑𝑅 Er 𝑋)
32ersymb 6120 . . . . . . . 8 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
43biimpa 280 . . . . . . 7 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51, 4jca 290 . . . . . 6 ((𝜑𝐴𝑅𝐵) → (𝜑𝐵𝑅𝐴))
62ertr 6121 . . . . . . 7 (𝜑 → ((𝐵𝑅𝐴𝐴𝑅𝑥) → 𝐵𝑅𝑥))
76impl 362 . . . . . 6 (((𝜑𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
85, 7sylan 267 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
92ertr 6121 . . . . . 6 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝑥) → 𝐴𝑅𝑥))
109impl 362 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
118, 10impbida 528 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝐴𝑅𝑥𝐵𝑅𝑥))
12 vex 2560 . . . . 5 𝑥 ∈ V
13 erth.2 . . . . . 6 (𝜑𝐴𝑋)
1413adantr 261 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐴𝑋)
15 elecg 6144 . . . . 5 ((𝑥 ∈ V ∧ 𝐴𝑋) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
1612, 14, 15sylancr 393 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
17 errel 6115 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
182, 17syl 14 . . . . . 6 (𝜑 → Rel 𝑅)
19 brrelex2 4383 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
2018, 19sylan 267 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐵 ∈ V)
21 elecg 6144 . . . . 5 ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2212, 20, 21sylancr 393 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2311, 16, 223bitr4d 209 . . 3 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑅))
2423eqrdv 2038 . 2 ((𝜑𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
252adantr 261 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝑅 Er 𝑋)
262, 13erref 6126 . . . . . . 7 (𝜑𝐴𝑅𝐴)
2726adantr 261 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
2813adantr 261 . . . . . . 7 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑋)
29 elecg 6144 . . . . . . 7 ((𝐴𝑋𝐴𝑋) → (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 2116 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
3425, 32ereldm 6149 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴𝑋𝐵𝑋))
3528, 34mpbid 135 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑋)
36 elecg 6144 . . . . 5 ((𝐴𝑋𝐵𝑋) → (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 6118 . 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 1243  wcel 1393  Vcvv 2557   class class class wbr 3764  Rel wrel 4350   Er wer 6103  [cec 6104
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-er 6106  df-ec 6108
This theorem is referenced by:  erth2  6151  erthi  6152  qliftfun  6188  eroveu  6197  th3qlem1  6208  enqeceq  6455  enq0eceq  6533  nnnq0lem1  6542  enreceq  6819  prsrlem1  6825
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