Theorem ordintdif 6413

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 3011	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) ≠ ∅)
3		dfdif2 3922	. . . 4 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
4	3	inteqi 4920	. . 3 ⊢ ∩ (𝐴 ∖ 𝐵) = ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
5		ordtri1 6395	. . . . . 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 6383	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
9		ordtri1 6395	. . . . . . . . . . 11 ⊢ ((Ord 𝐵 ∧ Ord 𝑥) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
10	7, 8, 9	syl2anr 608	. . . . . . . . . 10 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ Ord 𝐵) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
11	10	an32s 664	. . . . . . . . 9 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐵 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐵))
12	11	rabbidva 3429	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵})
13	12	inteqd 4921	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵})
14		intmin 4937	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = 𝐵)
15	13, 14	sylan9req 2825	. . . . . 6 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝐵 ∈ 𝐴) → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵)
16	15	ex 417	. . . . 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 1133	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴 ⊆ 𝐵) → ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = 𝐵)
19	4, 18	eqtr2id 2817	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 = ∩ (𝐴 ∖ 𝐵))
20	2, 19	syl3an3br 1433	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → 𝐵 = ∩ (𝐴 ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  {crab 3423   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  ∩ cint 4916  Ord word 6360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364

This theorem is referenced by: (None)