Theorem unblem1 9322

Description: Lemma for unbnn 9326. 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 7915	. . . . . 6 ⊢ ω ⊆ On
2		sstr 3921	. . . . . 6 ⊢ ((𝐵 ⊆ ω ∧ ω ⊆ On) → 𝐵 ⊆ On)
3	1, 2	mpan2 704	. . . . 5 ⊢ (𝐵 ⊆ ω → 𝐵 ⊆ On)
4	3	ssdifssd 4086	. . . 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 3907	. . . . . 6 ⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → 𝐴 ∈ ω))
7		peano2b 7928	. . . . . 6 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
8	6, 7	imbitrdi 254	. . . . 5 ⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ ω))
9		eleq1 2852	. . . . . . . 8 ⊢ (𝑥 = suc 𝐴 → (𝑥 ∈ 𝑦 ↔ suc 𝐴 ∈ 𝑦))
10	9	rexbidv 3190	. . . . . . 7 ⊢ (𝑥 = suc 𝐴 → (∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ↔ ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦))
11	10	rspccva 3579	. . . . . 6 ⊢ ((∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦)
12		ssel 3907	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → 𝑦 ∈ ω))
13		nnord 7919	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → Ord 𝑦)
14		ordn2lp 6408	. . . . . . . . . . . . . 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 4052	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))
22		ne0i 4290	. . . . . . . . . 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 3165	. . . . . 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 7838	. . 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 4054	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 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  ∩ cint 4912  Ord word 6387  Oncon0 6388  suc csuc 6390  ωcom 7911

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-om 7912

This theorem is used by:  unblem2  9323  unblem3  9324