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Theorem bnj986 31532
Description: Technical lemma for bnj69 31586. 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 31306 . . . . . 6 (𝑛𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
41, 3bnj769 31340 . . . . 5 (𝜒 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
54bnj1196 31373 . . . 4 (𝜒 → ∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚))
6 vex 3386 . . . . . 6 𝑛 ∈ V
76sucex 7243 . . . . 5 suc 𝑛 ∈ V
87isseti 3395 . . . 4 𝑝 𝑝 = suc 𝑛
95, 8jctir 517 . . 3 (𝜒 → (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
10 exdistr 2050 . . . 4 (∃𝑚𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛) ↔ ∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
11 19.41v 2045 . . . 4 (∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
1210, 11bitr2i 268 . . 3 ((∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ ∃𝑚𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
139, 12sylib 210 . 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 1110 . . . 4 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
1614, 15bitri 267 . . 3 (𝜏 ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
17162exbii 1945 . 2 (∃𝑚𝑝𝜏 ↔ ∃𝑚𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
1813, 17sylibr 226 1 (𝜒 → ∃𝑚𝑝𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  wrex 3088  cdif 3764  c0 4113  {csn 4366  suc csuc 5941   Fn wfn 6094  ωcom 7297  w-bnj17 31263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-tr 4944  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-om 7298  df-bnj17 31264
This theorem is referenced by:  bnj996  31533
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