Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj556 Structured version   Visualization version   GIF version

Theorem bnj556 30944
Description: Technical lemma for bnj852 30965. 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
bnj556.18 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj556.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
Assertion
Ref Expression
bnj556 (𝜂𝜎)

Proof of Theorem bnj556
StepHypRef Expression
1 vex 3198 . . . . 5 𝑝 ∈ V
21bnj216 30774 . . . 4 (𝑚 = suc 𝑝𝑝𝑚)
323anim3i 1248 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) → (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
43adantr 481 . 2 (((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω) → (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
5 bnj556.19 . . 3 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
6 bnj258 30748 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
75, 6bitri 264 . 2 (𝜂 ↔ ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
8 bnj556.18 . 2 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
94, 7, 83imtr4i 281 1 (𝜂𝜎)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  suc csuc 5713  ωcom 7050  w-bnj17 30726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-un 3572  df-sn 4169  df-suc 5717  df-bnj17 30727
This theorem is referenced by:  bnj557  30945  bnj561  30947  bnj562  30948
  Copyright terms: Public domain W3C validator