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Theorem erdisj 8782
Description: Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by NM, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
erdisj (𝑅 Er 𝑋 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))

Proof of Theorem erdisj
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 neq0 4347 . . . 4 (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅))
2 simpl 481 . . . . . . 7 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑅 Er 𝑋)
3 elinel1 4195 . . . . . . . . . 10 (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐴]𝑅)
43adantl 480 . . . . . . . . 9 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐴]𝑅)
5 vex 3475 . . . . . . . . . 10 𝑥 ∈ V
6 ecexr 8734 . . . . . . . . . . 11 (𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
74, 6syl 17 . . . . . . . . . 10 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴 ∈ V)
8 elecg 8772 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
95, 7, 8sylancr 585 . . . . . . . . 9 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
104, 9mpbid 231 . . . . . . . 8 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝑥)
11 elinel2 4196 . . . . . . . . . 10 (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐵]𝑅)
1211adantl 480 . . . . . . . . 9 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐵]𝑅)
13 ecexr 8734 . . . . . . . . . . 11 (𝑥 ∈ [𝐵]𝑅𝐵 ∈ V)
1412, 13syl 17 . . . . . . . . . 10 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵 ∈ V)
15 elecg 8772 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
165, 14, 15sylancr 585 . . . . . . . . 9 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
1712, 16mpbid 231 . . . . . . . 8 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵𝑅𝑥)
182, 10, 17ertr4d 8748 . . . . . . 7 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝐵)
192, 18erthi 8781 . . . . . 6 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → [𝐴]𝑅 = [𝐵]𝑅)
2019ex 411 . . . . 5 (𝑅 Er 𝑋 → (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
2120exlimdv 1928 . . . 4 (𝑅 Er 𝑋 → (∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
221, 21biimtrid 241 . . 3 (𝑅 Er 𝑋 → (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ → [𝐴]𝑅 = [𝐵]𝑅))
2322orrd 861 . 2 (𝑅 Er 𝑋 → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ [𝐴]𝑅 = [𝐵]𝑅))
2423orcomd 869 1 (𝑅 Er 𝑋 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wex 1773  wcel 2098  Vcvv 3471  cin 3946  c0 4324   class class class wbr 5150   Er wer 8726  [cec 8727
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-er 8729  df-ec 8731
This theorem is referenced by:  qsdisj  8817
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