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Theorem eqvreldisj 35841
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.) (Revised by Peter Mazsa, 2-Jun-2019.)
Assertion
Ref Expression
eqvreldisj ( EqvRel 𝑅 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))

Proof of Theorem eqvreldisj
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 neq0 4307 . . . 4 (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅))
2 simpl 485 . . . . . . 7 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → EqvRel 𝑅)
3 elinel1 4170 . . . . . . . . . 10 (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐴]𝑅)
43adantl 484 . . . . . . . . 9 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐴]𝑅)
5 ecexr 8286 . . . . . . . . . . 11 (𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
64, 5syl 17 . . . . . . . . . 10 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴 ∈ V)
7 vex 3496 . . . . . . . . . 10 𝑥 ∈ V
8 elecALTV 35519 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
96, 7, 8sylancl 588 . . . . . . . . 9 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
104, 9mpbid 234 . . . . . . . 8 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝑥)
11 elinel2 4171 . . . . . . . . . 10 (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐵]𝑅)
1211adantl 484 . . . . . . . . 9 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐵]𝑅)
13 ecexr 8286 . . . . . . . . . . 11 (𝑥 ∈ [𝐵]𝑅𝐵 ∈ V)
1412, 13syl 17 . . . . . . . . . 10 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵 ∈ V)
15 elecALTV 35519 . . . . . . . . . 10 ((𝐵 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
1614, 7, 15sylancl 588 . . . . . . . . 9 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
1712, 16mpbid 234 . . . . . . . 8 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵𝑅𝑥)
182, 10, 17eqvreltr4d 35836 . . . . . . 7 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝐵)
192, 18eqvrelthi 35840 . . . . . 6 (( EqvRel 𝑅𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → [𝐴]𝑅 = [𝐵]𝑅)
2019ex 415 . . . . 5 ( EqvRel 𝑅 → (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
2120exlimdv 1928 . . . 4 ( EqvRel 𝑅 → (∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
221, 21syl5bi 244 . . 3 ( EqvRel 𝑅 → (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ → [𝐴]𝑅 = [𝐵]𝑅))
2322orrd 859 . 2 ( EqvRel 𝑅 → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ [𝐴]𝑅 = [𝐵]𝑅))
2423orcomd 867 1 ( EqvRel 𝑅 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1531  wex 1774  wcel 2108  Vcvv 3493  cin 3933  c0 4289   class class class wbr 5057  [cec 8279   EqvRel weqvrel 35462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8283  df-refrel 35744  df-symrel 35772  df-trrel 35802  df-eqvrel 35812
This theorem is referenced by:  qsdisjALTV  35842
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