Theorem disjnf 30322

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 4197	. . . 4 ⊢ (𝐵 ∩ 𝐵) = 𝐵
2	1	eqeq1i 2828	. . 3 ⊢ ((𝐵 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅)
3	2	orbi1i 910	. 2 ⊢ (((𝐵 ∩ 𝐵) = ∅ ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦) ↔ (𝐵 = ∅ ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
4		eqidd 2824	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐵)
5	4	disjor 5048	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐵) = ∅))
6		orcom 866	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐵) = ∅) ↔ ((𝐵 ∩ 𝐵) = ∅ ∨ 𝑥 = 𝑦))
7	6	ralbii 3167	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐵) = ∅) ↔ ∀𝑦 ∈ 𝐴 ((𝐵 ∩ 𝐵) = ∅ ∨ 𝑥 = 𝑦))
8		r19.32v 3342	. . . . 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 3167	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐵) = ∅) ↔ ∀𝑥 ∈ 𝐴 ((𝐵 ∩ 𝐵) = ∅ ∨ ∀𝑦 ∈ 𝐴 𝑥 = 𝑦))
11		r19.32v 3342	. . 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 30254	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)
14	13	orbi2i 909	. 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 843   = wceq 1537   ∈ wcel 2114  ∃*wmo 2620  ∀wral 3140   ∩ cin 3937  ∅c0 4293  Disj wdisj 5033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rmo 3148  df-v 3498  df-dif 3941  df-in 3945  df-nul 4294  df-disj 5034

This theorem is referenced by: (None)