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Theorem unblem1 8253
Description: Lemma for unbnn 8257. 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 7111 . . . . . 6 ω ⊆ On
2 sstr 3644 . . . . . 6 ((𝐵 ⊆ ω ∧ ω ⊆ On) → 𝐵 ⊆ On)
31, 2mpan2 707 . . . . 5 (𝐵 ⊆ ω → 𝐵 ⊆ On)
43ssdifssd 3781 . . . 4 (𝐵 ⊆ ω → (𝐵 ∖ suc 𝐴) ⊆ On)
54ad2antrr 762 . . 3 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ⊆ On)
6 ssel 3630 . . . . . 6 (𝐵 ⊆ ω → (𝐴𝐵𝐴 ∈ ω))
7 peano2b 7123 . . . . . 6 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
86, 7syl6ib 241 . . . . 5 (𝐵 ⊆ ω → (𝐴𝐵 → suc 𝐴 ∈ ω))
9 eleq1 2718 . . . . . . . 8 (𝑥 = suc 𝐴 → (𝑥𝑦 ↔ suc 𝐴𝑦))
109rexbidv 3081 . . . . . . 7 (𝑥 = suc 𝐴 → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 suc 𝐴𝑦))
1110rspccva 3339 . . . . . 6 ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦𝐵 suc 𝐴𝑦)
12 ssel 3630 . . . . . . . . . . 11 (𝐵 ⊆ ω → (𝑦𝐵𝑦 ∈ ω))
13 nnord 7115 . . . . . . . . . . . 12 (𝑦 ∈ ω → Ord 𝑦)
14 ordn2lp 5781 . . . . . . . . . . . . . 14 (Ord 𝑦 → ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴𝑦))
15 imnan 437 . . . . . . . . . . . . . 14 ((𝑦 ∈ suc 𝐴 → ¬ suc 𝐴𝑦) ↔ ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴𝑦))
1614, 15sylibr 224 . . . . . . . . . . . . 13 (Ord 𝑦 → (𝑦 ∈ suc 𝐴 → ¬ suc 𝐴𝑦))
1716con2d 129 . . . . . . . . . . . 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 728 . . . . . . . . 9 (𝐵 ⊆ ω → ((𝑦𝐵 ∧ suc 𝐴𝑦) → (𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴)))
21 eldif 3617 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))
22 ne0i 3954 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∖ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
2321, 22sylbir 225 . . . . . . . . 9 ((𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
2420, 23syl6 35 . . . . . . . 8 (𝐵 ⊆ ω → ((𝑦𝐵 ∧ suc 𝐴𝑦) → (𝐵 ∖ suc 𝐴) ≠ ∅))
2524expd 451 . . . . . . 7 (𝐵 ⊆ ω → (𝑦𝐵 → (suc 𝐴𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅)))
2625rexlimdv 3059 . . . . . 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 498 . . . 4 (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦𝐴𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅))
2928impl 649 . . 3 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅)
30 onint 7037 . . 3 (((𝐵 ∖ suc 𝐴) ⊆ On ∧ (𝐵 ∖ suc 𝐴) ≠ ∅) → (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
315, 29, 30syl2anc 694 . 2 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
3231eldifad 3619 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 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  cdif 3604  wss 3607  c0 3948   cint 4507  Ord word 5760  Oncon0 5761  suc csuc 5763  ωcom 7107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-om 7108
This theorem is referenced by:  unblem2  8254  unblem3  8255
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