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Theorem unblem1 9325
Description: Lemma for unbnn 9329. 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 ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ∈ 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem unblem1
StepHypRef Expression
1 omsson 7890 . . . . . 6 ω ⊆ On
2 sstr 4003 . . . . . 6 ((𝐵 ⊆ ω ∧ ω ⊆ On) → 𝐵 ⊆ On)
31, 2mpan2 691 . . . . 5 (𝐵 ⊆ ω → 𝐵 ⊆ On)
43ssdifssd 4156 . . . 4 (𝐵 ⊆ ω → (𝐵 ∖ suc 𝐴) ⊆ On)
54ad2antrr 726 . . 3 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ⊆ On)
6 ssel 3988 . . . . . 6 (𝐵 ⊆ ω → (𝐴𝐵𝐴 ∈ ω))
7 peano2b 7903 . . . . . 6 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
86, 7imbitrdi 251 . . . . 5 (𝐵 ⊆ ω → (𝐴𝐵 → suc 𝐴 ∈ ω))
9 eleq1 2826 . . . . . . . 8 (𝑥 = suc 𝐴 → (𝑥𝑦 ↔ suc 𝐴𝑦))
109rexbidv 3176 . . . . . . 7 (𝑥 = suc 𝐴 → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 suc 𝐴𝑦))
1110rspccva 3620 . . . . . 6 ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦𝐵 suc 𝐴𝑦)
12 ssel 3988 . . . . . . . . . . 11 (𝐵 ⊆ ω → (𝑦𝐵𝑦 ∈ ω))
13 nnord 7894 . . . . . . . . . . . 12 (𝑦 ∈ ω → Ord 𝑦)
14 ordn2lp 6405 . . . . . . . . . . . . . 14 (Ord 𝑦 → ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴𝑦))
15 imnan 399 . . . . . . . . . . . . . 14 ((𝑦 ∈ suc 𝐴 → ¬ suc 𝐴𝑦) ↔ ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴𝑦))
1614, 15sylibr 234 . . . . . . . . . . . . 13 (Ord 𝑦 → (𝑦 ∈ suc 𝐴 → ¬ suc 𝐴𝑦))
1716con2d 134 . . . . . . . . . . . 12 (Ord 𝑦 → (suc 𝐴𝑦 → ¬ 𝑦 ∈ suc 𝐴))
1813, 17syl 17 . . . . . . . . . . 11 (𝑦 ∈ ω → (suc 𝐴𝑦 → ¬ 𝑦 ∈ suc 𝐴))
1912, 18syl6 35 . . . . . . . . . 10 (𝐵 ⊆ ω → (𝑦𝐵 → (suc 𝐴𝑦 → ¬ 𝑦 ∈ suc 𝐴)))
2019imdistand 570 . . . . . . . . 9 (𝐵 ⊆ ω → ((𝑦𝐵 ∧ suc 𝐴𝑦) → (𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴)))
21 eldif 3972 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))
22 ne0i 4346 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∖ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
2321, 22sylbir 235 . . . . . . . . 9 ((𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
2420, 23syl6 35 . . . . . . . 8 (𝐵 ⊆ ω → ((𝑦𝐵 ∧ suc 𝐴𝑦) → (𝐵 ∖ suc 𝐴) ≠ ∅))
2524expd 415 . . . . . . 7 (𝐵 ⊆ ω → (𝑦𝐵 → (suc 𝐴𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅)))
2625rexlimdv 3150 . . . . . 6 (𝐵 ⊆ ω → (∃𝑦𝐵 suc 𝐴𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅))
2711, 26syl5 34 . . . . 5 (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦 ∧ suc 𝐴 ∈ ω) → (𝐵 ∖ suc 𝐴) ≠ ∅))
288, 27sylan2d 605 . . . 4 (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦𝐴𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅))
2928impl 455 . . 3 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅)
30 onint 7809 . . 3 (((𝐵 ∖ suc 𝐴) ⊆ On ∧ (𝐵 ∖ suc 𝐴) ≠ ∅) → (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
315, 29, 30syl2anc 584 . 2 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
3231eldifad 3974 1 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  cdif 3959  wss 3962  c0 4338   cint 4950  Ord word 6384  Oncon0 6385  suc csuc 6387  ωcom 7886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-om 7887
This theorem is referenced by:  unblem2  9326  unblem3  9327
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