Theorem unblem1 9356

Description: Lemma for unbnn 9360. 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 7907	. . . . . 6 ⊢ ω ⊆ On
2		sstr 4017	. . . . . 6 ⊢ ((𝐵 ⊆ ω ∧ ω ⊆ On) → 𝐵 ⊆ On)
3	1, 2	mpan2 690	. . . . 5 ⊢ (𝐵 ⊆ ω → 𝐵 ⊆ On)
4	3	ssdifssd 4170	. . . 4 ⊢ (𝐵 ⊆ ω → (𝐵 ∖ suc 𝐴) ⊆ On)
5	4	ad2antrr 725	. . 3 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ⊆ On)
6		ssel 4002	. . . . . 6 ⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → 𝐴 ∈ ω))
7		peano2b 7920	. . . . . 6 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
8	6, 7	imbitrdi 251	. . . . 5 ⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ ω))
9		eleq1 2832	. . . . . . . 8 ⊢ (𝑥 = suc 𝐴 → (𝑥 ∈ 𝑦 ↔ suc 𝐴 ∈ 𝑦))
10	9	rexbidv 3185	. . . . . . 7 ⊢ (𝑥 = suc 𝐴 → (∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ↔ ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦))
11	10	rspccva 3634	. . . . . 6 ⊢ ((∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦)
12		ssel 4002	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → 𝑦 ∈ ω))
13		nnord 7911	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → Ord 𝑦)
14		ordn2lp 6415	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑦 → ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝑦))
15		imnan 399	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ suc 𝐴 → ¬ suc 𝐴 ∈ 𝑦) ↔ ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝑦))
16	14, 15	sylibr 234	. . . . . . . . . . . . 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 570	. . . . . . . . 9 ⊢ (𝐵 ⊆ ω → ((𝑦 ∈ 𝐵 ∧ suc 𝐴 ∈ 𝑦) → (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴)))
21		eldif 3986	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))
22		ne0i 4364	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
23	21, 22	sylbir 235	. . . . . . . . 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 415	. . . . . . 7 ⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → (suc 𝐴 ∈ 𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅)))
26	25	rexlimdv 3159	. . . . . 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 604	. . . 4 ⊢ (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅))
29	28	impl 455	. . 3 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅)
30		onint 7826	. . 3 ⊢ (((𝐵 ∖ suc 𝐴) ⊆ On ∧ (𝐵 ∖ suc 𝐴) ≠ ∅) → ∩ (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
31	5, 29, 30	syl2anc 583	. 2 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
32	31	eldifad 3988	1 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  ∩ cint 4970  Ord word 6394  Oncon0 6395  suc csuc 6397  ωcom 7903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-om 7904

This theorem is referenced by:  unblem2  9357  unblem3  9358