Theorem erdisj 8813

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.

Step	Hyp	Ref	Expression
1		neq0 4302	. . . 4 ⊢ (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅))
2		simpl 488	. . . . . . 7 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑅 Er 𝑋)
3		elinel1 4101	. . . . . . . . . 10 ⊢ (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐴]𝑅)
4	3	adantl 487	. . . . . . . . 9 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐴]𝑅)
5		vex 3458	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
6		ecexr 8760	. . . . . . . . . . 11 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
7	4, 6	syl 18	. . . . . . . . . 10 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴 ∈ V)
8		elecg 8800	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
9	5, 7, 8	sylancr 599	. . . . . . . . 9 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
10	4, 9	mpbid 235	. . . . . . . 8 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝑥)
11		elinel2 4102	. . . . . . . . . 10 ⊢ (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐵]𝑅)
12	11	adantl 487	. . . . . . . . 9 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐵]𝑅)
13		ecexr 8760	. . . . . . . . . . 11 ⊢ (𝑥 ∈ [𝐵]𝑅 → 𝐵 ∈ V)
14	12, 13	syl 18	. . . . . . . . . 10 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵 ∈ V)
15		elecg 8800	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
16	5, 14, 15	sylancr 599	. . . . . . . . 9 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
17	12, 16	mpbid 235	. . . . . . . 8 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵𝑅𝑥)
18	2, 10, 17	ertr4d 8775	. . . . . . 7 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝐵)
19	2, 18	erthi 8812	. . . . . 6 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → [𝐴]𝑅 = [𝐵]𝑅)
20	19	ex 418	. . . . 5 ⊢ (𝑅 Er 𝑋 → (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
21	20	exlimdv 1968	. . . 4 ⊢ (𝑅 Er 𝑋 → (∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
22	1, 21	biimtrid 245	. . 3 ⊢ (𝑅 Er 𝑋 → (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ → [𝐴]𝑅 = [𝐵]𝑅))
23	22	orrd 877	. 2 ⊢ (𝑅 Er 𝑋 → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ [𝐴]𝑅 = [𝐵]𝑅))
24	23	orcomd 885	1 ⊢ (𝑅 Er 𝑋 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092  ∅c0 4281   class class class wbr 5108   Er wer 8752  [cec 8753

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-er 8755  df-ec 8757

This theorem is used by:  qsdisj  8853