Theorem disjnf 32770

Description: In case 𝑥 is not free in 𝐵, disjointness is not so interesting since it reduces to cases where 𝐴 is a singleton. (Google Groups discussion with Peter Mazsa.) (Contributed by Thierry Arnoux, 26-Jul-2018.)

Assertion

Ref	Expression
disjnf	⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ (𝐵 = ∅ ∨ ∃*𝑥 𝑥 ∈ 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disjnf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inidm 4178	. . . 4 ⊢ (𝐵 ∩ 𝐵) = 𝐵
2	1	eqeq1i 2767	. . 3 ⊢ ((𝐵 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅)
3	2	orbi1i 924	. 2 ⊢ (((𝐵 ∩ 𝐵) = ∅ ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦) ↔ (𝐵 = ∅ ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
4		eqidd 2763	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐵)
5	4	disjor 5082	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐵) = ∅))
6		orcom 881	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐵) = ∅) ↔ ((𝐵 ∩ 𝐵) = ∅ ∨ 𝑥 = 𝑦))
7	6	ralbii 3108	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐵) = ∅) ↔ ∀𝑦 ∈ 𝐴 ((𝐵 ∩ 𝐵) = ∅ ∨ 𝑥 = 𝑦))
8		r19.32v 3195	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 ((𝐵 ∩ 𝐵) = ∅ ∨ 𝑥 = 𝑦) ↔ ((𝐵 ∩ 𝐵) = ∅ ∨ ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
9	7, 8	bitri 277	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐵) = ∅) ↔ ((𝐵 ∩ 𝐵) = ∅ ∨ ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
10	9	ralbii 3108	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐵) = ∅) ↔ ∀𝑥 ∈ 𝐴 ((𝐵 ∩ 𝐵) = ∅ ∨ ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
11		r19.32v 3195	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ((𝐵 ∩ 𝐵) = ∅ ∨ ∀𝑦 ∈ 𝐴 𝑥 = 𝑦) ↔ ((𝐵 ∩ 𝐵) = ∅ ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
12	5, 10, 11	3bitri 299	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ((𝐵 ∩ 𝐵) = ∅ ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
13		moel 3387	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)
14	13	orbi2i 923	. 2 ⊢ ((𝐵 = ∅ ∨ ∃*𝑥 𝑥 ∈ 𝐴) ↔ (𝐵 = ∅ ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
15	3, 12, 14	3bitr4i 305	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ (𝐵 = ∅ ∨ ∃*𝑥 𝑥 ∈ 𝐴))

Syntax hints:   ↔ wb 208   ∨ wo 858   = wceq 1560   ∈ wcel 2142  ∃*wmo 2564  ∀wral 3076   ∩ cin 3903  ∅c0 4285  Disj wdisj 5067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rmo 3367  df-v 3456  df-dif 3907  df-in 3911  df-nul 4286  df-disj 5068

This theorem is referenced by: (None)