Theorem erdisj 8550

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 4279	. . . 4 ⊢ (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅))
2		simpl 483	. . . . . . 7 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑅 Er 𝑋)
3		elinel1 4129	. . . . . . . . . 10 ⊢ (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐴]𝑅)
4	3	adantl 482	. . . . . . . . 9 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐴]𝑅)
5		vex 3436	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
6		ecexr 8503	. . . . . . . . . . 11 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
7	4, 6	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴 ∈ V)
8		elecg 8541	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
9	5, 7, 8	sylancr 587	. . . . . . . . 9 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
10	4, 9	mpbid 231	. . . . . . . 8 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝑥)
11		elinel2 4130	. . . . . . . . . 10 ⊢ (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐵]𝑅)
12	11	adantl 482	. . . . . . . . 9 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐵]𝑅)
13		ecexr 8503	. . . . . . . . . . 11 ⊢ (𝑥 ∈ [𝐵]𝑅 → 𝐵 ∈ V)
14	12, 13	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵 ∈ V)
15		elecg 8541	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
16	5, 14, 15	sylancr 587	. . . . . . . . 9 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
17	12, 16	mpbid 231	. . . . . . . 8 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵𝑅𝑥)
18	2, 10, 17	ertr4d 8517	. . . . . . 7 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝐵)
19	2, 18	erthi 8549	. . . . . 6 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → [𝐴]𝑅 = [𝐵]𝑅)
20	19	ex 413	. . . . 5 ⊢ (𝑅 Er 𝑋 → (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
21	20	exlimdv 1936	. . . 4 ⊢ (𝑅 Er 𝑋 → (∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
22	1, 21	syl5bi 241	. . 3 ⊢ (𝑅 Er 𝑋 → (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ → [𝐴]𝑅 = [𝐵]𝑅))
23	22	orrd 860	. 2 ⊢ (𝑅 Er 𝑋 → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ [𝐴]𝑅 = [𝐵]𝑅))
24	23	orcomd 868	1 ⊢ (𝑅 Er 𝑋 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886  ∅c0 4256   class class class wbr 5074   Er wer 8495  [cec 8496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-er 8498  df-ec 8500

This theorem is referenced by:  qsdisj  8583