Theorem unblem1 9259

Description: Lemma for unbnn 9263. 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 7872	. . . . . 6 ⊢ ω ⊆ On
2		sstr 3946	. . . . . 6 ⊢ ((𝐵 ⊆ ω ∧ ω ⊆ On) → 𝐵 ⊆ On)
3	1, 2	mpan2 704	. . . . 5 ⊢ (𝐵 ⊆ ω → 𝐵 ⊆ On)
4	3	ssdifssd 4101	. . . 4 ⊢ (𝐵 ⊆ ω → (𝐵 ∖ suc 𝐴) ⊆ On)
5	4	ad2antrr 739	. . 3 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ⊆ On)
6		ssel 3932	. . . . . 6 ⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → 𝐴 ∈ ω))
7		peano2b 7885	. . . . . 6 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
8	6, 7	imbitrdi 254	. . . . 5 ⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ ω))
9		eleq1 2853	. . . . . . . 8 ⊢ (𝑥 = suc 𝐴 → (𝑥 ∈ 𝑦 ↔ suc 𝐴 ∈ 𝑦))
10	9	rexbidv 3191	. . . . . . 7 ⊢ (𝑥 = suc 𝐴 → (∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ↔ ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦))
11	10	rspccva 3582	. . . . . 6 ⊢ ((∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦)
12		ssel 3932	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → 𝑦 ∈ ω))
13		nnord 7876	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → Ord 𝑦)
14		ordn2lp 6384	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑦 → ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝑦))
15		imnan 405	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ suc 𝐴 → ¬ suc 𝐴 ∈ 𝑦) ↔ ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝑦))
16	14, 15	sylibr 237	. . . . . . . . . . . . 13 ⊢ (Ord 𝑦 → (𝑦 ∈ suc 𝐴 → ¬ suc 𝐴 ∈ 𝑦))
17	16	con2d 135	. . . . . . . . . . . 12 ⊢ (Ord 𝑦 → (suc 𝐴 ∈ 𝑦 → ¬ 𝑦 ∈ suc 𝐴))
18	13, 17	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (suc 𝐴 ∈ 𝑦 → ¬ 𝑦 ∈ suc 𝐴))
19	12, 18	syl6 36	. . . . . . . . . 10 ⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → (suc 𝐴 ∈ 𝑦 → ¬ 𝑦 ∈ suc 𝐴)))
20	19	imdistand 581	. . . . . . . . 9 ⊢ (𝐵 ⊆ ω → ((𝑦 ∈ 𝐵 ∧ suc 𝐴 ∈ 𝑦) → (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴)))
21		eldif 3916	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))
22		ne0i 4294	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
23	21, 22	sylbir 238	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
24	20, 23	syl6 36	. . . . . . . 8 ⊢ (𝐵 ⊆ ω → ((𝑦 ∈ 𝐵 ∧ suc 𝐴 ∈ 𝑦) → (𝐵 ∖ suc 𝐴) ≠ ∅))
25	24	expd 421	. . . . . . 7 ⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → (suc 𝐴 ∈ 𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅)))
26	25	rexlimdv 3166	. . . . . 6 ⊢ (𝐵 ⊆ ω → (∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅))
27	11, 26	syl5 35	. . . . 5 ⊢ (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ suc 𝐴 ∈ ω) → (𝐵 ∖ suc 𝐴) ≠ ∅))
28	8, 27	sylan2d 617	. . . 4 ⊢ (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅))
29	28	impl 461	. . 3 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅)
30		onint 7795	. . 3 ⊢ (((𝐵 ∖ suc 𝐴) ⊆ On ∧ (𝐵 ∖ suc 𝐴) ≠ ∅) → ∩ (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
31	5, 29, 30	syl2anc 596	. 2 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
32	31	eldifad 3918	1 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4286  ∩ cint 4914  Ord word 6363  Oncon0 6364  suc csuc 6366  ωcom 7868

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-om 7869

This theorem is used by:  unblem2  9260  unblem3  9261