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Theorem unblem1 8074
Description: Lemma for unbnn 8078. 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 6938 . . . . . 6 ω ⊆ On
2 sstr 3575 . . . . . 6 ((𝐵 ⊆ ω ∧ ω ⊆ On) → 𝐵 ⊆ On)
31, 2mpan2 702 . . . . 5 (𝐵 ⊆ ω → 𝐵 ⊆ On)
43ssdifssd 3709 . . . 4 (𝐵 ⊆ ω → (𝐵 ∖ suc 𝐴) ⊆ On)
54ad2antrr 757 . . 3 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ⊆ On)
6 ssel 3561 . . . . . 6 (𝐵 ⊆ ω → (𝐴𝐵𝐴 ∈ ω))
7 peano2b 6950 . . . . . 6 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
86, 7syl6ib 239 . . . . 5 (𝐵 ⊆ ω → (𝐴𝐵 → suc 𝐴 ∈ ω))
9 eleq1 2675 . . . . . . . 8 (𝑥 = suc 𝐴 → (𝑥𝑦 ↔ suc 𝐴𝑦))
109rexbidv 3033 . . . . . . 7 (𝑥 = suc 𝐴 → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 suc 𝐴𝑦))
1110rspccva 3280 . . . . . 6 ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦𝐵 suc 𝐴𝑦)
12 ssel 3561 . . . . . . . . . . 11 (𝐵 ⊆ ω → (𝑦𝐵𝑦 ∈ ω))
13 nnord 6942 . . . . . . . . . . . 12 (𝑦 ∈ ω → Ord 𝑦)
14 ordn2lp 5646 . . . . . . . . . . . . . 14 (Ord 𝑦 → ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴𝑦))
15 imnan 436 . . . . . . . . . . . . . 14 ((𝑦 ∈ suc 𝐴 → ¬ suc 𝐴𝑦) ↔ ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴𝑦))
1614, 15sylibr 222 . . . . . . . . . . . . 13 (Ord 𝑦 → (𝑦 ∈ suc 𝐴 → ¬ suc 𝐴𝑦))
1716con2d 127 . . . . . . . . . . . 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 723 . . . . . . . . 9 (𝐵 ⊆ ω → ((𝑦𝐵 ∧ suc 𝐴𝑦) → (𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴)))
21 eldif 3549 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))
22 ne0i 3879 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∖ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
2321, 22sylbir 223 . . . . . . . . 9 ((𝑦𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅)
2420, 23syl6 34 . . . . . . . 8 (𝐵 ⊆ ω → ((𝑦𝐵 ∧ suc 𝐴𝑦) → (𝐵 ∖ suc 𝐴) ≠ ∅))
2524expd 450 . . . . . . 7 (𝐵 ⊆ ω → (𝑦𝐵 → (suc 𝐴𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅)))
2625rexlimdv 3011 . . . . . 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 497 . . . 4 (𝐵 ⊆ ω → ((∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦𝐴𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅))
2928impl 647 . . 3 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅)
30 onint 6864 . . 3 (((𝐵 ∖ suc 𝐴) ⊆ On ∧ (𝐵 ∖ suc 𝐴) ≠ ∅) → (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
315, 29, 30syl2anc 690 . 2 (((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴))
3231eldifad 3551 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 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  cdif 3536  wss 3539  c0 3873   cint 4404  Ord word 5625  Oncon0 5626  suc csuc 5628  ωcom 6934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-om 6935
This theorem is referenced by:  unblem2  8075  unblem3  8076
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