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Theorem bnj98 32247
Description: Technical lemma for bnj150 32256. 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 3447 . . . . . 6 𝑖 ∈ V
21sucid 6242 . . . . 5 𝑖 ∈ suc 𝑖
32n0ii 4255 . . . 4 ¬ suc 𝑖 = ∅
4 df-suc 6169 . . . . . 6 suc 𝑖 = (𝑖 ∪ {𝑖})
5 df-un 3889 . . . . . 6 (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
64, 5eqtri 2824 . . . . 5 suc 𝑖 = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
7 df1o2 8103 . . . . . . 7 1o = {∅}
86, 7eleq12i 2885 . . . . . 6 (suc 𝑖 ∈ 1o ↔ {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅})
9 elsni 4545 . . . . . 6 ({𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
108, 9sylbi 220 . . . . 5 (suc 𝑖 ∈ 1o → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
116, 10syl5eq 2848 . . . 4 (suc 𝑖 ∈ 1o → suc 𝑖 = ∅)
123, 11mto 200 . . 3 ¬ suc 𝑖 ∈ 1o
1312pm2.21i 119 . 2 (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
1413rgenw 3121 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 844   = wceq 1538  wcel 2112  {cab 2779  wral 3109  cun 3882  c0 4246  {csn 4528   ciun 4884  suc csuc 6165  cfv 6328  ωcom 7564  1oc1o 8082   predc-bnj14 32066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-nul 4247  df-sn 4529  df-suc 6169  df-1o 8089
This theorem is referenced by:  bnj150  32256
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