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Theorem bnj98 34712
Description: Technical lemma for bnj150 34721. 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.)
Assertion
Ref Expression
bnj98 𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))

Proof of Theorem bnj98
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3466 . . . . . 6 𝑖 ∈ V
21sucid 6458 . . . . 5 𝑖 ∈ suc 𝑖
32n0ii 4339 . . . 4 ¬ suc 𝑖 = ∅
4 df-suc 6382 . . . . . 6 suc 𝑖 = (𝑖 ∪ {𝑖})
5 df-un 3952 . . . . . 6 (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
64, 5eqtri 2754 . . . . 5 suc 𝑖 = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
7 df1o2 8503 . . . . . . 7 1o = {∅}
86, 7eleq12i 2819 . . . . . 6 (suc 𝑖 ∈ 1o ↔ {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅})
9 elsni 4650 . . . . . 6 ({𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
108, 9sylbi 216 . . . . 5 (suc 𝑖 ∈ 1o → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
116, 10eqtrid 2778 . . . 4 (suc 𝑖 ∈ 1o → suc 𝑖 = ∅)
123, 11mto 196 . . 3 ¬ suc 𝑖 ∈ 1o
1312pm2.21i 119 . 2 (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
1413rgenw 3055 1 𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 845   = wceq 1534  wcel 2099  {cab 2703  wral 3051  cun 3945  c0 4325  {csn 4633   ciun 5001  suc csuc 6378  cfv 6554  ωcom 7876  1oc1o 8489   predc-bnj14 34533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-v 3464  df-dif 3950  df-un 3952  df-nul 4326  df-sn 4634  df-suc 6382  df-1o 8496
This theorem is referenced by:  bnj150  34721
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