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Theorem erdisj 8372
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 4234 . . . 4 (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅))
2 simpl 486 . . . . . . 7 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑅 Er 𝑋)
3 elinel1 4085 . . . . . . . . . 10 (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐴]𝑅)
43adantl 485 . . . . . . . . 9 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐴]𝑅)
5 vex 3402 . . . . . . . . . 10 𝑥 ∈ V
6 ecexr 8325 . . . . . . . . . . 11 (𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
74, 6syl 17 . . . . . . . . . 10 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴 ∈ V)
8 elecg 8363 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
95, 7, 8sylancr 590 . . . . . . . . 9 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
104, 9mpbid 235 . . . . . . . 8 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝑥)
11 elinel2 4086 . . . . . . . . . 10 (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐵]𝑅)
1211adantl 485 . . . . . . . . 9 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐵]𝑅)
13 ecexr 8325 . . . . . . . . . . 11 (𝑥 ∈ [𝐵]𝑅𝐵 ∈ V)
1412, 13syl 17 . . . . . . . . . 10 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵 ∈ V)
15 elecg 8363 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
165, 14, 15sylancr 590 . . . . . . . . 9 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
1712, 16mpbid 235 . . . . . . . 8 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵𝑅𝑥)
182, 10, 17ertr4d 8339 . . . . . . 7 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝐵)
192, 18erthi 8371 . . . . . 6 ((𝑅 Er 𝑋𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → [𝐴]𝑅 = [𝐵]𝑅)
2019ex 416 . . . . 5 (𝑅 Er 𝑋 → (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
2120exlimdv 1940 . . . 4 (𝑅 Er 𝑋 → (∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
221, 21syl5bi 245 . . 3 (𝑅 Er 𝑋 → (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ → [𝐴]𝑅 = [𝐵]𝑅))
2322orrd 862 . 2 (𝑅 Er 𝑋 → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ [𝐴]𝑅 = [𝐵]𝑅))
2423orcomd 870 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 209  wa 399  wo 846   = wceq 1542  wex 1786  wcel 2114  Vcvv 3398  cin 3842  c0 4211   class class class wbr 5030   Er wer 8317  [cec 8318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-er 8320  df-ec 8322
This theorem is referenced by:  qsdisj  8405
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