Theorem ncdisjeq 6148

Description: Two cardinals are either disjoint or equal. (Contributed by SF, 25-Feb-2015.)

Assertion

Ref	Expression
ncdisjeq	⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A ∩ B) = ∅ ∨ A = B))

Proof of Theorem ncdisjeq

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

Step	Hyp	Ref	Expression
1		elncs 6119	. . . 4 ⊢ (A ∈ NC ↔ ∃x A = Nc x)
2		elncs 6119	. . . 4 ⊢ (B ∈ NC ↔ ∃y B = Nc y)
3	1, 2	anbi12i 678	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ) ↔ (∃x A = Nc x ∧ ∃y B = Nc y))
4		eeanv 1913	. . 3 ⊢ (∃x∃y(A = Nc x ∧ B = Nc y) ↔ (∃x A = Nc x ∧ ∃y B = Nc y))
5	3, 4	bitr4i 243	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ) ↔ ∃x∃y(A = Nc x ∧ B = Nc y))
6		ener 6039	. . . . . 6 ⊢ ≈ Er V
7		erdisj 5972	. . . . . 6 ⊢ ( ≈ Er V → ([x] ≈ = [y] ≈ ∨ ([x] ≈ ∩ [y] ≈ ) = ∅))
8	6, 7	ax-mp 5	. . . . 5 ⊢ ([x] ≈ = [y] ≈ ∨ ([x] ≈ ∩ [y] ≈ ) = ∅)
9		df-nc 6101	. . . . . . 7 ⊢ Nc x = [x] ≈
10		eqtr 2370	. . . . . . 7 ⊢ ((A = Nc x ∧ Nc x = [x] ≈ ) → A = [x] ≈ )
11	9, 10	mpan2 652	. . . . . 6 ⊢ (A = Nc x → A = [x] ≈ )
12		df-nc 6101	. . . . . . 7 ⊢ Nc y = [y] ≈
13		eqtr 2370	. . . . . . 7 ⊢ ((B = Nc y ∧ Nc y = [y] ≈ ) → B = [y] ≈ )
14	12, 13	mpan2 652	. . . . . 6 ⊢ (B = Nc y → B = [y] ≈ )
15		eqeq12 2365	. . . . . . 7 ⊢ ((A = [x] ≈ ∧ B = [y] ≈ ) → (A = B ↔ [x] ≈ = [y] ≈ ))
16		ineq12 3452	. . . . . . . 8 ⊢ ((A = [x] ≈ ∧ B = [y] ≈ ) → (A ∩ B) = ([x] ≈ ∩ [y] ≈ ))
17	16	eqeq1d 2361	. . . . . . 7 ⊢ ((A = [x] ≈ ∧ B = [y] ≈ ) → ((A ∩ B) = ∅ ↔ ([x] ≈ ∩ [y] ≈ ) = ∅))
18	15, 17	orbi12d 690	. . . . . 6 ⊢ ((A = [x] ≈ ∧ B = [y] ≈ ) → ((A = B ∨ (A ∩ B) = ∅) ↔ ([x] ≈ = [y] ≈ ∨ ([x] ≈ ∩ [y] ≈ ) = ∅)))
19	11, 14, 18	syl2an 463	. . . . 5 ⊢ ((A = Nc x ∧ B = Nc y) → ((A = B ∨ (A ∩ B) = ∅) ↔ ([x] ≈ = [y] ≈ ∨ ([x] ≈ ∩ [y] ≈ ) = ∅)))
20	8, 19	mpbiri 224	. . . 4 ⊢ ((A = Nc x ∧ B = Nc y) → (A = B ∨ (A ∩ B) = ∅))
21	20	orcomd 377	. . 3 ⊢ ((A = Nc x ∧ B = Nc y) → ((A ∩ B) = ∅ ∨ A = B))
22	21	exlimivv 1635	. 2 ⊢ (∃x∃y(A = Nc x ∧ B = Nc y) → ((A ∩ B) = ∅ ∨ A = B))
23	5, 22	sylbi 187	1 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A ∩ B) = ∅ ∨ A = B))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208  ∅c0 3550   class class class wbr 4639   Er cer 5898  [cec 5945   ≈ cen 6028   NC cncs 6088   Nc cnc 6091

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101

This theorem is referenced by:  nceleq  6149