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Theorem unblem1 9328
Description: Lemma for unbnn 9332. 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 7891 . . . . . 6 ω ⊆ On
2 sstr 3992 . . . . . 6 ((𝐵 ⊆ ω ∧ ω ⊆ On) → 𝐵 ⊆ On)
31, 2mpan2 691 . . . . 5 (𝐵 ⊆ ω → 𝐵 ⊆ On)
43ssdifssd 4147 . . . 4 (𝐵 ⊆ ω → (𝐵 ∖ suc 𝐴) ⊆ On)
54ad2antrr 726 . . 3 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ⊆ On)
6 ssel 3977 . . . . . 6 (𝐵 ⊆ ω → (𝐴𝐵𝐴 ∈ ω))
7 peano2b 7904 . . . . . 6 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
86, 7imbitrdi 251 . . . . 5 (𝐵 ⊆ ω → (𝐴𝐵 → suc 𝐴 ∈ ω))
9 eleq1 2829 . . . . . . . 8 (𝑥 = suc 𝐴 → (𝑥𝑦 ↔ suc 𝐴𝑦))
109rexbidv 3179 . . . . . . 7 (𝑥 = suc 𝐴 → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 suc 𝐴𝑦))
1110rspccva 3621 . . . . . 6 ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦𝐵 suc 𝐴𝑦)
12 ssel 3977 . . . . . . . . . . 11 (𝐵 ⊆ ω → (𝑦𝐵𝑦 ∈ ω))
13 nnord 7895 . . . . . . . . . . . 12 (𝑦 ∈ ω → Ord 𝑦)
14 ordn2lp 6404 . . . . . . . . . . . . . 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 3961 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))
22 ne0i 4341 . . . . . . . . . 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 3153 . . . . . 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 7810 . . 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 3963 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 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cdif 3948  wss 3951  c0 4333   cint 4946  Ord word 6383  Oncon0 6384  suc csuc 6386  ωcom 7887
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-om 7888
This theorem is referenced by:  unblem2  9329  unblem3  9330
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