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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 8	. . . . 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))

Colors of variables: wff setvar class

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-1 5 ax-2 6 ax-3 7 ax-mp 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

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