Theorem unblem1 9291

Description: Lemma for unbnn 9295. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set. (Contributed by NM, 3-Dec-2003.)

Assertion

Ref	Expression
unblem1	⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem unblem1

Step	Hyp	Ref	Expression
1		omsson 7855	. . . . . 6 ⊢ ω ⊆ On
2		sstr 3989	. . . . . 6 ⊢ ((𝐵 ⊆ ω ∧ ω ⊆ On) → 𝐵 ⊆ On)
3	1, 2	mpan2 689	. . . . 5 ⊢ (𝐵 ⊆ ω → 𝐵 ⊆ On)
4	3	ssdifssd 4141	. . . 4 ⊢ (𝐵 ⊆ ω → (𝐵 ∖ suc 𝐴) ⊆ On)
5	4	ad2antrr 724	. . 3 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ⊆ On)
6		ssel 3974	. . . . . 6 ⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → 𝐴 ∈ ω))
7		peano2b 7868	. . . . . 6 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
8	6, 7	imbitrdi 250	. . . . 5 ⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ ω))
9		eleq1 2821	. . . . . . . 8 ⊢ (𝑥 = suc 𝐴 → (𝑥 ∈ 𝑦 ↔ suc 𝐴 ∈ 𝑦))
10	9	rexbidv 3178	. . . . . . 7 ⊢ (𝑥 = suc 𝐴 → (∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ↔ ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦))
11	10	rspccva 3611	. . . . . 6 ⊢ ((∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦)
12		ssel 3974	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → 𝑦 ∈ ω))
13		nnord 7859	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → Ord 𝑦)
14		ordn2lp 6381	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑦 → ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝑦))
15		imnan 400	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ suc 𝐴 → ¬ suc 𝐴 ∈ 𝑦) ↔ ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝑦))
16	14, 15	sylibr 233	. . . . . . . . . . . . 13 ⊢ (Ord 𝑦 → (𝑦 ∈ suc 𝐴 → ¬ suc 𝐴 ∈ 𝑦))
17	16	con2d 134	. . . . . . . . . . . 12 ⊢ (Ord 𝑦 → (suc 𝐴 ∈ 𝑦 → ¬ 𝑦 ∈ suc 𝐴))
18	13, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (suc 𝐴 ∈ 𝑦 → ¬ 𝑦 ∈ suc 𝐴))
19	12, 18	syl6 35	. . . . . . . . . 10 ⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → (suc 𝐴 ∈ 𝑦 → ¬ 𝑦 ∈ suc 𝐴)))
20	19	imdistand 571	. . . . . . . . 9 ⊢ (𝐵 ⊆ ω → ((𝑦 ∈ 𝐵 ∧ suc 𝐴 ∈ 𝑦) → (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴)))
21		eldif 3957	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))
22		ne0i 4333	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
23	21, 22	sylbir 234	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
24	20, 23	syl6 35	. . . . . . . 8 ⊢ (𝐵 ⊆ ω → ((𝑦 ∈ 𝐵 ∧ suc 𝐴 ∈ 𝑦) → (𝐵 ∖ suc 𝐴) ≠ ∅))
25	24	expd 416	. . . . . . 7 ⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → (suc 𝐴 ∈ 𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅)))
26	25	rexlimdv 3153	. . . . . 6 ⊢ (𝐵 ⊆ ω → (∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅))
27	11, 26	syl5 34	. . . . 5 ⊢ (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ suc 𝐴 ∈ ω) → (𝐵 ∖ suc 𝐴) ≠ ∅))
28	8, 27	sylan2d 605	. . . 4 ⊢ (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅))
29	28	impl 456	. . 3 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅)
30		onint 7774	. . 3 ⊢ (((𝐵 ∖ suc 𝐴) ⊆ On ∧ (𝐵 ∖ suc 𝐴) ≠ ∅) → ∩ (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
31	5, 29, 30	syl2anc 584	. 2 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
32	31	eldifad 3959	1 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∖ cdif 3944   ⊆ wss 3947  ∅c0 4321  ∩ cint 4949  Ord word 6360  Oncon0 6361  suc csuc 6363  ωcom 7851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-om 7852

This theorem is referenced by:  unblem2  9292  unblem3  9293