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Theorem bnj986 31009
 Description: Technical lemma for bnj69 31063. 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 30782 . . . . . 6 (𝑛𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
41, 3bnj769 30817 . . . . 5 (𝜒 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
54bnj1196 30850 . . . 4 (𝜒 → ∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚))
6 vex 3201 . . . . . 6 𝑛 ∈ V
76sucex 7008 . . . . 5 suc 𝑛 ∈ V
87isseti 3207 . . . 4 𝑝 𝑝 = suc 𝑛
95, 8jctir 561 . . 3 (𝜒 → (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
10 exdistr 1918 . . . 4 (∃𝑚𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛) ↔ ∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
11 19.41v 1913 . . . 4 (∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
1210, 11bitr2i 265 . . 3 ((∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ ∃𝑚𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
139, 12sylib 208 . 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 1039 . . . 4 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
1614, 15bitri 264 . . 3 (𝜏 ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
17162exbii 1774 . 2 (∃𝑚𝑝𝜏 ↔ ∃𝑚𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
1813, 17sylibr 224 1 (𝜒 → ∃𝑚𝑝𝜏)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1482  ∃wex 1703   ∈ wcel 1989  ∃wrex 2912   ∖ cdif 3569  ∅c0 3913  {csn 4175  suc csuc 5723   Fn wfn 5881  ωcom 7062   ∧ w-bnj17 30737 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-tr 4751  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-om 7063  df-bnj17 30738 This theorem is referenced by:  bnj996  31010
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