Theorem disjdifprg 28770

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 4576	. . . . . 6 ⊢ Disj 𝑥 ∈ {∅}𝑥
2		simpr 476	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → 𝐵 = ∅)
3		eqidd 2611	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → ∅ = ∅)
4		elex 3185	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
5		0ex 4718	. . . . . . . . . . 11 ⊢ ∅ ∈ V
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → ∅ ∈ V)
7	4, 6, 6	preqsnd 4330	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑊 → ({𝐵, ∅} = {∅} ↔ (𝐵 = ∅ ∧ ∅ = ∅)))
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → ({𝐵, ∅} = {∅} ↔ (𝐵 = ∅ ∧ ∅ = ∅)))
9	2, 3, 8	mpbir2and 959	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → {𝐵, ∅} = {∅})
10	9	disjeq1d 4561	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ Disj 𝑥 ∈ {∅}𝑥))
11	1, 10	mpbiri 247	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 = ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
12		in0 3920	. . . . . 6 ⊢ (𝐵 ∩ ∅) = ∅
13	4	adantr 480	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
14	5	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → ∅ ∈ V)
15		simpr 476	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
17		id 22	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
18	16, 17	disjprg 4578	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ ∅ ∈ V ∧ 𝐵 ≠ ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ (𝐵 ∩ ∅) = ∅))
19	13, 14, 15, 18	syl3anc 1318	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → (Disj 𝑥 ∈ {𝐵, ∅}𝑥 ↔ (𝐵 ∩ ∅) = ∅))
20	12, 19	mpbiri 247	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
21	11, 20	pm2.61dane 2869	. . . 4 ⊢ (𝐵 ∈ 𝑊 → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
22	21	ad2antlr 759	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → Disj 𝑥 ∈ {𝐵, ∅}𝑥)
23		difeq2 3684	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅))
24		dif0 3904	. . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵
25	23, 24	syl6eq 2660	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = 𝐵)
26		id 22	. . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
27	25, 26	preq12d 4220	. . . . 5 ⊢ (𝐴 = ∅ → {(𝐵 ∖ 𝐴), 𝐴} = {𝐵, ∅})
28	27	disjeq1d 4561	. . . 4 ⊢ (𝐴 = ∅ → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ Disj 𝑥 ∈ {𝐵, ∅}𝑥))
29	28	adantl 481	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ Disj 𝑥 ∈ {𝐵, ∅}𝑥))
30	22, 29	mpbird 246	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = ∅) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)
31		incom 3767	. . . 4 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∩ 𝐴)
32		disjdif 3992	. . . 4 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
33	31, 32	eqtr3i 2634	. . 3 ⊢ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅
34		difexg 4735	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∖ 𝐴) ∈ V)
35	34	ad2antlr 759	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ∈ V)
36		elex 3185	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
37	36	ad2antrr 758	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → 𝐴 ∈ V)
38		ssid 3587	. . . . . 6 ⊢ (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴)
39		ssdifeq0 4003	. . . . . . . 8 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
40	39	notbii 309	. . . . . . 7 ⊢ (¬ 𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ¬ 𝐴 = ∅)
41		nssne2 3625	. . . . . . 7 ⊢ (((𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴) ∧ ¬ 𝐴 ⊆ (𝐵 ∖ 𝐴)) → (𝐵 ∖ 𝐴) ≠ 𝐴)
42	40, 41	sylan2br 492	. . . . . 6 ⊢ (((𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ 𝐴) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ≠ 𝐴)
43	38, 42	mpan 702	. . . . 5 ⊢ (¬ 𝐴 = ∅ → (𝐵 ∖ 𝐴) ≠ 𝐴)
44	43	adantl 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (𝐵 ∖ 𝐴) ≠ 𝐴)
45		id 22	. . . . 5 ⊢ (𝑥 = (𝐵 ∖ 𝐴) → 𝑥 = (𝐵 ∖ 𝐴))
46		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
47	45, 46	disjprg 4578	. . . 4 ⊢ (((𝐵 ∖ 𝐴) ∈ V ∧ 𝐴 ∈ V ∧ (𝐵 ∖ 𝐴) ≠ 𝐴) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅))
48	35, 37, 44, 47	syl3anc 1318	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → (Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥 ↔ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅))
49	33, 48	mpbiri 247	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ¬ 𝐴 = ∅) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)
50	30, 49	pm2.61dan 828	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  Disj wdisj 4553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-disj 4554

This theorem is referenced by:  disjdifprg2  28771  measssd  29605