Theorem qsdisj 5996

Description: Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)

Hypotheses

Ref	Expression
qsdisj.1	⊢ (φ → R Er V)
qsdisj.2	⊢ (φ → B ∈ (A / R))
qsdisj.3	⊢ (φ → C ∈ (A / R))

Assertion

Ref	Expression
qsdisj	⊢ (φ → (B = C ∨ (B ∩ C) = ∅))

Proof of Theorem qsdisj

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qsdisj.2	. 2 ⊢ (φ → B ∈ (A / R))
2		eqid 2353	. . 3 ⊢ (A / R) = (A / R)
3		eqeq1 2359	. . . 4 ⊢ ([x]R = B → ([x]R = C ↔ B = C))
4		ineq1 3451	. . . . 5 ⊢ ([x]R = B → ([x]R ∩ C) = (B ∩ C))
5	4	eqeq1d 2361	. . . 4 ⊢ ([x]R = B → (([x]R ∩ C) = ∅ ↔ (B ∩ C) = ∅))
6	3, 5	orbi12d 690	. . 3 ⊢ ([x]R = B → (([x]R = C ∨ ([x]R ∩ C) = ∅) ↔ (B = C ∨ (B ∩ C) = ∅)))
7		qsdisj.3	. . . . 5 ⊢ (φ → C ∈ (A / R))
8	7	adantr 451	. . . 4 ⊢ ((φ ∧ x ∈ A) → C ∈ (A / R))
9		eqeq2 2362	. . . . . 6 ⊢ ([y]R = C → ([x]R = [y]R ↔ [x]R = C))
10		ineq2 3452	. . . . . . 7 ⊢ ([y]R = C → ([x]R ∩ [y]R) = ([x]R ∩ C))
11	10	eqeq1d 2361	. . . . . 6 ⊢ ([y]R = C → (([x]R ∩ [y]R) = ∅ ↔ ([x]R ∩ C) = ∅))
12	9, 11	orbi12d 690	. . . . 5 ⊢ ([y]R = C → (([x]R = [y]R ∨ ([x]R ∩ [y]R) = ∅) ↔ ([x]R = C ∨ ([x]R ∩ C) = ∅)))
13		qsdisj.1	. . . . . . 7 ⊢ (φ → R Er V)
14	13	ad2antrr 706	. . . . . 6 ⊢ (((φ ∧ x ∈ A) ∧ y ∈ A) → R Er V)
15		erdisj 5973	. . . . . 6 ⊢ (R Er V → ([x]R = [y]R ∨ ([x]R ∩ [y]R) = ∅))
16	14, 15	syl 15	. . . . 5 ⊢ (((φ ∧ x ∈ A) ∧ y ∈ A) → ([x]R = [y]R ∨ ([x]R ∩ [y]R) = ∅))
17	2, 12, 16	ectocld 5992	. . . 4 ⊢ (((φ ∧ x ∈ A) ∧ C ∈ (A / R)) → ([x]R = C ∨ ([x]R ∩ C) = ∅))
18	8, 17	mpdan 649	. . 3 ⊢ ((φ ∧ x ∈ A) → ([x]R = C ∨ ([x]R ∩ C) = ∅))
19	2, 6, 18	ectocld 5992	. 2 ⊢ ((φ ∧ B ∈ (A / R)) → (B = C ∨ (B ∩ C) = ∅))
20	1, 19	mpdan 649	1 ⊢ (φ → (B = C ∨ (B ∩ C) = ∅))

Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∩ cin 3209  ∅c0 3551   class class class wbr 4640   Er cer 5899  [cec 5946   / cqs 5947

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  df-qs 5952

This theorem is referenced by: (None)