Theorem ordintdif 6414

Description: If 𝐵 is smaller than 𝐴, then it equals the intersection of the difference. Exercise 11 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)

Assertion

Ref	Expression
ordintdif	⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → 𝐵 = ∩ (𝐴 ∖ 𝐵))

Proof of Theorem ordintdif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssdif0 4363	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
2	1	necon3bbii 2987	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) ≠ ∅)
3		dfdif2 3957	. . . 4 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
4	3	inteqi 4954	. . 3 ⊢ ∩ (𝐴 ∖ 𝐵) = ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
5		ordtri1 6397	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))
6	5	con2bid 354	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 ↔ ¬ 𝐴 ⊆ 𝐵))
7		id 22	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → Ord 𝐵)
8		ordelord 6386	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
9		ordtri1 6397	. . . . . . . . . . 11 ⊢ ((Ord 𝐵 ∧ Ord 𝑥) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
10	7, 8, 9	syl2anr 596	. . . . . . . . . 10 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ Ord 𝐵) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
11	10	an32s 649	. . . . . . . . 9 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
12	11	rabbidva 3438	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵})
13	12	inteqd 4955	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵})
14		intmin 4972	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = 𝐵)
15	13, 14	sylan9req 2792	. . . . . 6 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝐵 ∈ 𝐴) → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵)
16	15	ex 412	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵))
17	6, 16	sylbird 260	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴 ⊆ 𝐵 → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵))
18	17	3impia 1116	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴 ⊆ 𝐵) → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵)
19	4, 18	eqtr2id 2784	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 = ∩ (𝐴 ∖ 𝐵))
20	2, 19	syl3an3br 1407	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → 𝐵 = ∩ (𝐴 ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  {crab 3431   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322  ∩ cint 4950  Ord word 6363

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367

This theorem is referenced by: (None)