Theorem ordintdif 6416

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 4321	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
2	1	necon3bbii 3007	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) ≠ ∅)
3		dfdif2 3915	. . . 4 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
4	3	inteqi 4918	. . 3 ⊢ ∩ (𝐴 ∖ 𝐵) = ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
5		ordtri1 6398	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))
6	5	con2bid 357	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 ↔ ¬ 𝐴 ⊆ 𝐵))
7		id 23	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → Ord 𝐵)
8		ordelord 6386	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
9		ordtri1 6398	. . . . . . . . . . 11 ⊢ ((Ord 𝐵 ∧ Ord 𝑥) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
10	7, 8, 9	syl2anr 609	. . . . . . . . . 10 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ Ord 𝐵) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
11	10	an32s 665	. . . . . . . . 9 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
12	11	rabbidva 3424	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵})
13	12	inteqd 4919	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵})
14		intmin 4935	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = 𝐵)
15	13, 14	sylan9req 2821	. . . . . 6 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝐵 ∈ 𝐴) → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵)
16	15	ex 418	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵))
17	6, 16	sylbird 263	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴 ⊆ 𝐵 → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵))
18	17	3impia 1135	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴 ⊆ 𝐵) → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵)
19	4, 18	eqtr2id 2813	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 = ∩ (𝐴 ∖ 𝐵))
20	2, 19	syl3an3br 1435	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → 𝐵 = ∩ (𝐴 ∖ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  {crab 3418   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4286  ∩ cint 4914  Ord word 6363

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 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367

This theorem is used by: (None)