Theorem erdisj 8508

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 4276	. . . 4 ⊢ (¬ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅))
2		simpl 482	. . . . . . 7 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑅 Er 𝑋)
3		elinel1 4125	. . . . . . . . . 10 ⊢ (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐴]𝑅)
4	3	adantl 481	. . . . . . . . 9 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐴]𝑅)
5		vex 3426	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
6		ecexr 8461	. . . . . . . . . . 11 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
7	4, 6	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴 ∈ V)
8		elecg 8499	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
9	5, 7, 8	sylancr 586	. . . . . . . . 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 4126	. . . . . . . . . 10 ⊢ (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → 𝑥 ∈ [𝐵]𝑅)
12	11	adantl 481	. . . . . . . . 9 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝑥 ∈ [𝐵]𝑅)
13		ecexr 8461	. . . . . . . . . . 11 ⊢ (𝑥 ∈ [𝐵]𝑅 → 𝐵 ∈ V)
14	12, 13	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐵 ∈ V)
15		elecg 8499	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
16	5, 14, 15	sylancr 586	. . . . . . . . 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 8475	. . . . . . 7 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → 𝐴𝑅𝐵)
19	2, 18	erthi 8507	. . . . . 6 ⊢ ((𝑅 Er 𝑋 ∧ 𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅)) → [𝐴]𝑅 = [𝐵]𝑅)
20	19	ex 412	. . . . 5 ⊢ (𝑅 Er 𝑋 → (𝑥 ∈ ([𝐴]𝑅 ∩ [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅))
21	20	exlimdv 1937	. . . 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 859	. 2 ⊢ (𝑅 Er 𝑋 → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ [𝐴]𝑅 = [𝐵]𝑅))
24	23	orcomd 867	1 ⊢ (𝑅 Er 𝑋 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882  ∅c0 4253   class class class wbr 5070   Er wer 8453  [cec 8454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-er 8456  df-ec 8458

This theorem is referenced by:  qsdisj  8541