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Theorem bnj986 34969
Description: Technical lemma for bnj69 35024. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj986.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj986.10 𝐷 = (ω ∖ {∅})
bnj986.15 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
Assertion
Ref Expression
bnj986 (𝜒 → ∃𝑚𝑝𝜏)
Distinct variable group:   𝑚,𝑛,𝑝
Allowed substitution hints:   𝜑(𝑓,𝑚,𝑛,𝑝)   𝜓(𝑓,𝑚,𝑛,𝑝)   𝜒(𝑓,𝑚,𝑛,𝑝)   𝜏(𝑓,𝑚,𝑛,𝑝)   𝐷(𝑓,𝑚,𝑛,𝑝)

Proof of Theorem bnj986
StepHypRef Expression
1 bnj986.3 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
2 bnj986.10 . . . . . . 7 𝐷 = (ω ∖ {∅})
32bnj158 34743 . . . . . 6 (𝑛𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
41, 3bnj769 34776 . . . . 5 (𝜒 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
54bnj1196 34808 . . . 4 (𝜒 → ∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚))
6 vex 3484 . . . . . 6 𝑛 ∈ V
76sucex 7826 . . . . 5 suc 𝑛 ∈ V
87isseti 3498 . . . 4 𝑝 𝑝 = suc 𝑛
95, 8jctir 520 . . 3 (𝜒 → (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
10 exdistr 1954 . . . 4 (∃𝑚𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛) ↔ ∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
11 19.41v 1949 . . . 4 (∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
1210, 11bitr2i 276 . . 3 ((∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ ∃𝑚𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
139, 12sylib 218 . 2 (𝜒 → ∃𝑚𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
14 bnj986.15 . . . 4 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
15 df-3an 1089 . . . 4 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
1614, 15bitri 275 . . 3 (𝜏 ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
17162exbii 1849 . 2 (∃𝑚𝑝𝜏 ↔ ∃𝑚𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
1813, 17sylibr 234 1 (𝜒 → ∃𝑚𝑝𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wrex 3070  cdif 3948  c0 4333  {csn 4626  suc csuc 6386   Fn wfn 6556  ωcom 7887  w-bnj17 34700
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-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  df-bnj17 34701
This theorem is referenced by:  bnj996  34970
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