Theorem erdisj 5973

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 set.mm contributors, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
erdisj	⊢ (R Er V → ([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 3561	. . . 4 ⊢ (¬ ([A]R ∩ [B]R) = ∅ ↔ ∃x x ∈ ([A]R ∩ [B]R))
2		simpl 443	. . . . . . 7 ⊢ ((R Er V ∧ x ∈ ([A]R ∩ [B]R)) → R Er V)
3		inss1 3476	. . . . . . . . . . 11 ⊢ ([A]R ∩ [B]R) ⊆ [A]R
4	3	sseli 3270	. . . . . . . . . 10 ⊢ (x ∈ ([A]R ∩ [B]R) → x ∈ [A]R)
5	4	adantl 452	. . . . . . . . 9 ⊢ ((R Er V ∧ x ∈ ([A]R ∩ [B]R)) → x ∈ [A]R)
6		ecexr 5951	. . . . . . . . 9 ⊢ (x ∈ [A]R → A ∈ V)
7	5, 6	syl 15	. . . . . . . 8 ⊢ ((R Er V ∧ x ∈ ([A]R ∩ [B]R)) → A ∈ V)
8		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
9	8	a1i 10	. . . . . . . 8 ⊢ ((R Er V ∧ x ∈ ([A]R ∩ [B]R)) → x ∈ V)
10		inss2 3477	. . . . . . . . . . 11 ⊢ ([A]R ∩ [B]R) ⊆ [B]R
11	10	sseli 3270	. . . . . . . . . 10 ⊢ (x ∈ ([A]R ∩ [B]R) → x ∈ [B]R)
12	11	adantl 452	. . . . . . . . 9 ⊢ ((R Er V ∧ x ∈ ([A]R ∩ [B]R)) → x ∈ [B]R)
13		ecexr 5951	. . . . . . . . 9 ⊢ (x ∈ [B]R → B ∈ V)
14	12, 13	syl 15	. . . . . . . 8 ⊢ ((R Er V ∧ x ∈ ([A]R ∩ [B]R)) → B ∈ V)
15		elec 5965	. . . . . . . . . 10 ⊢ (x ∈ [A]R ↔ ARx)
16	4, 15	sylib 188	. . . . . . . . 9 ⊢ (x ∈ ([A]R ∩ [B]R) → ARx)
17	16	adantl 452	. . . . . . . 8 ⊢ ((R Er V ∧ x ∈ ([A]R ∩ [B]R)) → ARx)
18		elec 5965	. . . . . . . . . 10 ⊢ (x ∈ [B]R ↔ BRx)
19	11, 18	sylib 188	. . . . . . . . 9 ⊢ (x ∈ ([A]R ∩ [B]R) → BRx)
20	19	adantl 452	. . . . . . . 8 ⊢ ((R Er V ∧ x ∈ ([A]R ∩ [B]R)) → BRx)
21	2, 7, 9, 14, 17, 20	ertr4d 5959	. . . . . . 7 ⊢ ((R Er V ∧ x ∈ ([A]R ∩ [B]R)) → ARB)
22	2, 21	erthi 5971	. . . . . 6 ⊢ ((R Er V ∧ x ∈ ([A]R ∩ [B]R)) → [A]R = [B]R)
23	22	ex 423	. . . . 5 ⊢ (R Er V → (x ∈ ([A]R ∩ [B]R) → [A]R = [B]R))
24	23	exlimdv 1636	. . . 4 ⊢ (R Er V → (∃x x ∈ ([A]R ∩ [B]R) → [A]R = [B]R))
25	1, 24	syl5bi 208	. . 3 ⊢ (R Er V → (¬ ([A]R ∩ [B]R) = ∅ → [A]R = [B]R))
26	25	orrd 367	. 2 ⊢ (R Er V → (([A]R ∩ [B]R) = ∅ ∨ [A]R = [B]R))
27	26	orcomd 377	1 ⊢ (R Er V → ([A]R = [B]R ∨ ([A]R ∩ [B]R) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∩ cin 3209  ∅c0 3551   class class class wbr 4640   Er cer 5899  [cec 5946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948

This theorem is referenced by:  qsdisj  5996  ncdisjeq  6149