Theorem disjdifprg 30914

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 5067	. . . . . 6 ⊢ Disj 𝑥 ∈ {∅}𝑥
2		simpr 485	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → 𝐵 = ∅)
3		eqidd 2739	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → ∅ = ∅)
4		id 22	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑊)
5		0ex 5231	. . . . . . . . . . 11 ⊢ ∅ ∈ V
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → ∅ ∈ V)
7	4, 6	preqsnd 4789	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑊 → ({𝐵, ∅} = {∅} ↔ (𝐵 = ∅ ∧ ∅ = ∅)))
8	7	adantr 481	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → ({𝐵, ∅} = {∅} ↔ (𝐵 = ∅ ∧ ∅ = ∅)))
9	2, 3, 8	mpbir2and 710	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → {𝐵, ∅} = {∅})
10	9	disjeq1d 5047	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ Disj 𝑥 ∈ {∅}𝑥))
11	1, 10	mpbiri 257	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
12		in0 4325	. . . . . 6 ⊢ (𝐵 ∩ ∅) = ∅
13		elex 3450	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
14	13	adantr 481	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
15	5	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → ∅ ∈ V)
16		simpr 485	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
17		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
18		id 22	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
19	17, 18	disjprg 5070	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ ∅ ∈ V ∧ 𝐵 ≠ ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ (𝐵 ∩ ∅) = ∅))
20	14, 15, 16, 19	syl3anc 1370	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ (𝐵 ∩ ∅) = ∅))
21	12, 20	mpbiri 257	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
22	11, 21	pm2.61dane 3032	. . . 4 ⊢ (𝐵 ∈ 𝑊 → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
23	22	ad2antlr 724	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
24		difeq2 4051	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅))
25		dif0 4306	. . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵
26	24, 25	eqtrdi 2794	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = 𝐵)
27		id 22	. . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
28	26, 27	preq12d 4677	. . . . 5 ⊢ (𝐴 = ∅ → {(𝐵 ∖ 𝐴), 𝐴} = {𝐵, ∅})
29	28	disjeq1d 5047	. . . 4 ⊢ (𝐴 = ∅ → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ Disj 𝑥 ∈ {𝐵, ∅}𝑥))
30	29	adantl 482	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ Disj 𝑥 ∈ {𝐵, ∅}𝑥))
31	23, 30	mpbird 256	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)
32		disjdifr 4406	. . 3 ⊢ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅
33		difexg 5251	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∖ 𝐴) ∈ V)
34	33	ad2antlr 724	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ∈ V)
35		elex 3450	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
36	35	ad2antrr 723	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → 𝐴 ∈ V)
37		ssid 3943	. . . . . 6 ⊢ (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴)
38		ssdifeq0 4417	. . . . . . . 8 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
39	38	notbii 320	. . . . . . 7 ⊢ (¬ 𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ¬ 𝐴 = ∅)
40		nssne2 3982	. . . . . . 7 ⊢ (((𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴) ∧ ¬ 𝐴 ⊆ (𝐵 ∖ 𝐴)) → (𝐵 ∖ 𝐴) ≠ 𝐴)
41	39, 40	sylan2br 595	. . . . . 6 ⊢ (((𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ≠ 𝐴)
42	37, 41	mpan 687	. . . . 5 ⊢ (¬ 𝐴 = ∅ → (𝐵 ∖ 𝐴) ≠ 𝐴)
43	42	adantl 482	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ≠ 𝐴)
44		id 22	. . . . 5 ⊢ (𝑥 = (𝐵 ∖ 𝐴) → 𝑥 = (𝐵 ∖ 𝐴))
45		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
46	44, 45	disjprg 5070	. . . 4 ⊢ (((𝐵 ∖ 𝐴) ∈ V ∧ 𝐴 ∈ V ∧ (𝐵 ∖ 𝐴) ≠ 𝐴) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅))
47	34, 36, 43, 46	syl3anc 1370	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅))
48	32, 47	mpbiri 257	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)
49	31, 48	pm2.61dan 810	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561  {cpr 4563  Disj wdisj 5039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rmo 3071  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-pr 4564  df-disj 5040

This theorem is referenced by:  disjdifprg2  30915  measssd  32183