Theorem unblem1 9215

Description: Lemma for unbnn 9219. 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 7826	. . . . . 6 ⊢ ω ⊆ On
2		sstr 3952	. . . . . 6 ⊢ ((𝐵 ⊆ ω ∧ ω ⊆ On) → 𝐵 ⊆ On)
3	1, 2	mpan2 691	. . . . 5 ⊢ (𝐵 ⊆ ω → 𝐵 ⊆ On)
4	3	ssdifssd 4106	. . . 4 ⊢ (𝐵 ⊆ ω → (𝐵 ∖ suc 𝐴) ⊆ On)
5	4	ad2antrr 726	. . 3 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ⊆ On)
6		ssel 3937	. . . . . 6 ⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → 𝐴 ∈ ω))
7		peano2b 7839	. . . . . 6 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
8	6, 7	imbitrdi 251	. . . . 5 ⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ ω))
9		eleq1 2816	. . . . . . . 8 ⊢ (𝑥 = suc 𝐴 → (𝑥 ∈ 𝑦 ↔ suc 𝐴 ∈ 𝑦))
10	9	rexbidv 3157	. . . . . . 7 ⊢ (𝑥 = suc 𝐴 → (∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ↔ ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦))
11	10	rspccva 3584	. . . . . 6 ⊢ ((∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦)
12		ssel 3937	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → 𝑦 ∈ ω))
13		nnord 7830	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → Ord 𝑦)
14		ordn2lp 6340	. . . . . . . . . . . . . 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 3921	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))
22		ne0i 4300	. . . . . . . . . 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 3132	. . . . . 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 455	. . 3 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅)
30		onint 7746	. . 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 3923	1 ⊢ (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ∖ cdif 3908   ⊆ wss 3911  ∅c0 4292  ∩ cint 4906  Ord word 6319  Oncon0 6320  suc csuc 6322  ωcom 7822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-om 7823

This theorem is referenced by:  unblem2  9216  unblem3  9217