Theorem disjdifprg 32597

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 5160	. . . . . 6 ⊢ Disj 𝑥 ∈ {∅}𝑥
2		simpr 484	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → 𝐵 = ∅)
3		eqidd 2741	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → ∅ = ∅)
4		id 22	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑊)
5		0ex 5325	. . . . . . . . . . 11 ⊢ ∅ ∈ V
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → ∅ ∈ V)
7	4, 6	preqsnd 4883	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑊 → ({𝐵, ∅} = {∅} ↔ (𝐵 = ∅ ∧ ∅ = ∅)))
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → ({𝐵, ∅} = {∅} ↔ (𝐵 = ∅ ∧ ∅ = ∅)))
9	2, 3, 8	mpbir2and 712	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → {𝐵, ∅} = {∅})
10	9	disjeq1d 5141	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ Disj 𝑥 ∈ {∅}𝑥))
11	1, 10	mpbiri 258	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
12		in0 4418	. . . . . 6 ⊢ (𝐵 ∩ ∅) = ∅
13		elex 3509	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
14	13	adantr 480	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
15	5	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → ∅ ∈ V)
16		simpr 484	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
17		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
18		id 22	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
19	17, 18	disjprg 5162	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ ∅ ∈ V ∧ 𝐵 ≠ ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ (𝐵 ∩ ∅) = ∅))
20	14, 15, 16, 19	syl3anc 1371	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ (𝐵 ∩ ∅) = ∅))
21	12, 20	mpbiri 258	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
22	11, 21	pm2.61dane 3035	. . . 4 ⊢ (𝐵 ∈ 𝑊 → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
23	22	ad2antlr 726	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
24		difeq2 4143	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅))
25		dif0 4400	. . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵
26	24, 25	eqtrdi 2796	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = 𝐵)
27		id 22	. . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
28	26, 27	preq12d 4766	. . . . 5 ⊢ (𝐴 = ∅ → {(𝐵 ∖ 𝐴), 𝐴} = {𝐵, ∅})
29	28	disjeq1d 5141	. . . 4 ⊢ (𝐴 = ∅ → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ Disj 𝑥 ∈ {𝐵, ∅}𝑥))
30	29	adantl 481	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ Disj 𝑥 ∈ {𝐵, ∅}𝑥))
31	23, 30	mpbird 257	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)
32		disjdifr 4496	. . 3 ⊢ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅
33		difexg 5347	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∖ 𝐴) ∈ V)
34	33	ad2antlr 726	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ∈ V)
35		elex 3509	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
36	35	ad2antrr 725	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → 𝐴 ∈ V)
37		ssid 4031	. . . . . 6 ⊢ (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴)
38		ssdifeq0 4510	. . . . . . . 8 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
39	38	notbii 320	. . . . . . 7 ⊢ (¬ 𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ¬ 𝐴 = ∅)
40		nssne2 4072	. . . . . . 7 ⊢ (((𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴) ∧ ¬ 𝐴 ⊆ (𝐵 ∖ 𝐴)) → (𝐵 ∖ 𝐴) ≠ 𝐴)
41	39, 40	sylan2br 594	. . . . . 6 ⊢ (((𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ≠ 𝐴)
42	37, 41	mpan 689	. . . . 5 ⊢ (¬ 𝐴 = ∅ → (𝐵 ∖ 𝐴) ≠ 𝐴)
43	42	adantl 481	. . . 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 5162	. . . 4 ⊢ (((𝐵 ∖ 𝐴) ∈ V ∧ 𝐴 ∈ V ∧ (𝐵 ∖ 𝐴) ≠ 𝐴) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅))
47	34, 36, 43, 46	syl3anc 1371	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅))
48	32, 47	mpbiri 258	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)
49	31, 48	pm2.61dan 812	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648  {cpr 4650  Disj wdisj 5133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-sn 4649  df-pr 4651  df-disj 5134

This theorem is referenced by:  disjdifprg2  32598  measssd  34179