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Theorem bnj98 33146
Description: Technical lemma for bnj150 33155. 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 3445 . . . . . 6 𝑖 ∈ V
21sucid 6383 . . . . 5 𝑖 ∈ suc 𝑖
32n0ii 4283 . . . 4 ¬ suc 𝑖 = ∅
4 df-suc 6308 . . . . . 6 suc 𝑖 = (𝑖 ∪ {𝑖})
5 df-un 3903 . . . . . 6 (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
64, 5eqtri 2764 . . . . 5 suc 𝑖 = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
7 df1o2 8374 . . . . . . 7 1o = {∅}
86, 7eleq12i 2829 . . . . . 6 (suc 𝑖 ∈ 1o ↔ {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅})
9 elsni 4590 . . . . . 6 ({𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
108, 9sylbi 216 . . . . 5 (suc 𝑖 ∈ 1o → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
116, 10eqtrid 2788 . . . 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 3065 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 1540  wcel 2105  {cab 2713  wral 3061  cun 3896  c0 4269  {csn 4573   ciun 4941  suc csuc 6304  cfv 6479  ωcom 7780  1oc1o 8360   predc-bnj14 32967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-v 3443  df-dif 3901  df-un 3903  df-nul 4270  df-sn 4574  df-suc 6308  df-1o 8367
This theorem is referenced by:  bnj150  33155
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