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Theorem unblem1 8097
 Description: Lemma for unbnn 8101. 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 6961 . . . . . 6 ω ⊆ On
2 sstr 3576 . . . . . 6 ((𝐵 ⊆ ω ∧ ω ⊆ On) → 𝐵 ⊆ On)
31, 2mpan2 703 . . . . 5 (𝐵 ⊆ ω → 𝐵 ⊆ On)
43ssdifssd 3710 . . . 4 (𝐵 ⊆ ω → (𝐵 ∖ suc 𝐴) ⊆ On)
54ad2antrr 758 . . 3 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ⊆ On)
6 ssel 3562 . . . . . 6 (𝐵 ⊆ ω → (𝐴𝐵𝐴 ∈ ω))
7 peano2b 6973 . . . . . 6 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
86, 7syl6ib 240 . . . . 5 (𝐵 ⊆ ω → (𝐴𝐵 → suc 𝐴 ∈ ω))
9 eleq1 2676 . . . . . . . 8 (𝑥 = suc 𝐴 → (𝑥𝑦 ↔ suc 𝐴𝑦))
109rexbidv 3034 . . . . . . 7 (𝑥 = suc 𝐴 → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 suc 𝐴𝑦))
1110rspccva 3281 . . . . . 6 ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦𝐵 suc 𝐴𝑦)
12 ssel 3562 . . . . . . . . . . 11 (𝐵 ⊆ ω → (𝑦𝐵𝑦 ∈ ω))
13 nnord 6965 . . . . . . . . . . . 12 (𝑦 ∈ ω → Ord 𝑦)
14 ordn2lp 5660 . . . . . . . . . . . . . 14 (Ord 𝑦 → ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴𝑦))
15 imnan 437 . . . . . . . . . . . . . 14 ((𝑦 ∈ suc 𝐴 → ¬ suc 𝐴𝑦) ↔ ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴𝑦))
1614, 15sylibr 223 . . . . . . . . . . . . 13 (Ord 𝑦 → (𝑦 ∈ suc 𝐴 → ¬ suc 𝐴𝑦))
1716con2d 128 . . . . . . . . . . . 12 (Ord 𝑦 → (suc 𝐴𝑦 → ¬ 𝑦 ∈ suc 𝐴))
1813, 17syl 17 . . . . . . . . . . 11 (𝑦 ∈ ω → (suc 𝐴𝑦 → ¬ 𝑦 ∈ suc 𝐴))
1912, 18syl6 34 . . . . . . . . . 10 (𝐵 ⊆ ω → (𝑦𝐵 → (suc 𝐴𝑦 → ¬ 𝑦 ∈ suc 𝐴)))
2019imdistand 724 . . . . . . . . 9 (𝐵 ⊆ ω → ((𝑦𝐵 ∧ suc 𝐴𝑦) → (𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴)))
21 eldif 3550 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))
22 ne0i 3880 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∖ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
2321, 22sylbir 224 . . . . . . . . 9 ((𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
2420, 23syl6 34 . . . . . . . 8 (𝐵 ⊆ ω → ((𝑦𝐵 ∧ suc 𝐴𝑦) → (𝐵 ∖ suc 𝐴) ≠ ∅))
2524expd 451 . . . . . . 7 (𝐵 ⊆ ω → (𝑦𝐵 → (suc 𝐴𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅)))
2625rexlimdv 3012 . . . . . 6 (𝐵 ⊆ ω → (∃𝑦𝐵 suc 𝐴𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅))
2711, 26syl5 33 . . . . 5 (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦 ∧ suc 𝐴 ∈ ω) → (𝐵 ∖ suc 𝐴) ≠ ∅))
288, 27sylan2d 498 . . . 4 (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦𝐴𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅))
2928impl 648 . . 3 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅)
30 onint 6887 . . 3 (((𝐵 ∖ suc 𝐴) ⊆ On ∧ (𝐵 ∖ suc 𝐴) ≠ ∅) → (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
315, 29, 30syl2anc 691 . 2 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
3231eldifad 3552 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 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  Ord word 5639  Oncon0 5640  suc csuc 5642  ωcom 6957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958 This theorem is referenced by:  unblem2  8098  unblem3  8099
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