Theorem disjdifprg 32950

Description: A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)

Assertion

Ref	Expression
disjdifprg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem disjdifprg

Step	Hyp	Ref	Expression
1		disjxsn 5108	. . . . . 6 ⊢ Disj 𝑥 ∈ {∅}𝑥
2		simpr 490	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → 𝐵 = ∅)
3		eqidd 2767	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → ∅ = ∅)
4		id 23	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑊)
5		0ex 5275	. . . . . . . . . . 11 ⊢ ∅ ∈ V
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → ∅ ∈ V)
7	4, 6	preqsnd 4829	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑊 → ({𝐵, ∅} = {∅} ↔ (𝐵 = ∅ ∧ ∅ = ∅)))
8	7	adantr 486	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → ({𝐵, ∅} = {∅} ↔ (𝐵 = ∅ ∧ ∅ = ∅)))
9	2, 3, 8	mpbir2and 726	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → {𝐵, ∅} = {∅})
10	9	disjeq1d 5089	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ Disj 𝑥 ∈ {∅}𝑥))
11	1, 10	mpbiri 261	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
12		in0 4355	. . . . . 6 ⊢ (𝐵 ∩ ∅) = ∅
13		elex 3479	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
14	13	adantr 486	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
15	5	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → ∅ ∈ V)
16		simpr 490	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
17		id 23	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
18		id 23	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
19	17, 18	disjprg 5110	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ ∅ ∈ V ∧ 𝐵 ≠ ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ (𝐵 ∩ ∅) = ∅))
20	14, 15, 16, 19	syl3anc 1398	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ (𝐵 ∩ ∅) = ∅))
21	12, 20	mpbiri 261	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
22	11, 21	pm2.61dane 3048	. . . 4 ⊢ (𝐵 ∈ 𝑊 → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
23	22	ad2antlr 740	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
24		difeq2 4078	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅))
25		dif0 4337	. . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵
26	24, 25	eqtrdi 2817	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = 𝐵)
27		id 23	. . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
28	26, 27	preq12d 4712	. . . . 5 ⊢ (𝐴 = ∅ → {(𝐵 ∖ 𝐴), 𝐴} = {𝐵, ∅})
29	28	disjeq1d 5089	. . . 4 ⊢ (𝐴 = ∅ → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ Disj 𝑥 ∈ {𝐵, ∅}𝑥))
30	29	adantl 487	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ Disj 𝑥 ∈ {𝐵, ∅}𝑥))
31	23, 30	mpbird 260	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)
32		disjdifr 4437	. . 3 ⊢ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅
33		difexg 5305	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∖ 𝐴) ∈ V)
34	33	ad2antlr 740	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ∈ V)
35		elex 3479	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
36	35	ad2antrr 739	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → 𝐴 ∈ V)
37		ssid 3962	. . . . . 6 ⊢ (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴)
38		ssdifeq0 4452	. . . . . . . 8 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
39	38	notbii 323	. . . . . . 7 ⊢ (¬ 𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ¬ 𝐴 = ∅)
40		nssne2 4003	. . . . . . 7 ⊢ (((𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴) ∧ ¬ 𝐴 ⊆ (𝐵 ∖ 𝐴)) → (𝐵 ∖ 𝐴) ≠ 𝐴)
41	39, 40	sylan2br 607	. . . . . 6 ⊢ (((𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ≠ 𝐴)
42	37, 41	mpan 703	. . . . 5 ⊢ (¬ 𝐴 = ∅ → (𝐵 ∖ 𝐴) ≠ 𝐴)
43	42	adantl 487	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ≠ 𝐴)
44		id 23	. . . . 5 ⊢ (𝑥 = (𝐵 ∖ 𝐴) → 𝑥 = (𝐵 ∖ 𝐴))
45		id 23	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
46	44, 45	disjprg 5110	. . . 4 ⊢ (((𝐵 ∖ 𝐴) ∈ V ∧ 𝐴 ∈ V ∧ (𝐵 ∖ 𝐴) ≠ 𝐴) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅))
47	34, 36, 43, 46	syl3anc 1398	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅))
48	32, 47	mpbiri 261	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)
49	31, 48	pm2.61dan 825	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  Vcvv 3458   ∖ cdif 3905   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  {csn 4594  {cpr 4596  Disj wdisj 5081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-sn 4595  df-pr 4597  df-disj 5082

This theorem is used by:  disjdifprg2  32951  measssd  34629