Theorem ordintdif 6417

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 4313	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
2	1	necon3bbii 3002	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) ≠ ∅)
3		dfdif2 4047	. . . 4 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
4	3	inteqi 4910	. . 3 ⊢ ∩ (𝐴 ∖ 𝐵) = ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
5		ordtri1 6399	. . . . . 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 6387	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
9		ordtri1 6399	. . . . . . . . . . 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 3418	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵})
13	12	inteqd 4911	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵})
14		intmin 4927	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ⊆ 𝑥} = 𝐵)
15	13, 14	sylan9req 2816	. . . . . 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 2808	. 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 1569   ∈ wcel 2144   ≠ wne 2955  {crab 3412   ⊆ wss 3896   ∖ cdif 4044  ∅c0 4277  ∩ cint 4906  Ord word 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6368

This theorem is used by: (None)