Theorem ordintdif 6383

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 4329	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
2	1	necon3bbii 2972	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) ≠ ∅)
3		dfdif2 3923	. . . 4 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
4	3	inteqi 4914	. . 3 ⊢ ∩ (𝐴 ∖ 𝐵) = ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
5		ordtri1 6365	. . . . . 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 6354	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
9		ordtri1 6365	. . . . . . . . . . 11 ⊢ ((Ord 𝐵 ∧ Ord 𝑥) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
10	7, 8, 9	syl2anr 597	. . . . . . . . . 10 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ Ord 𝐵) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
11	10	an32s 652	. . . . . . . . 9 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
12	11	rabbidva 3412	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵})
13	12	inteqd 4915	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵})
14		intmin 4932	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = 𝐵)
15	13, 14	sylan9req 2785	. . . . . 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 1117	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴 ⊆ 𝐵) → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵)
19	4, 18	eqtr2id 2777	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 = ∩ (𝐴 ∖ 𝐵))
20	2, 19	syl3an3br 1410	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → 𝐵 = ∩ (𝐴 ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3405   ∖ cdif 3911   ⊆ wss 3914  ∅c0 4296  ∩ cint 4910  Ord word 6331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335

This theorem is referenced by: (None)