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Theorem bnj98 30911
Description: Technical lemma for bnj150 30920. 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 3198 . . . . . 6 𝑖 ∈ V
21sucid 5792 . . . . 5 𝑖 ∈ suc 𝑖
32n0ii 3914 . . . 4 ¬ suc 𝑖 = ∅
4 df-suc 5717 . . . . . 6 suc 𝑖 = (𝑖 ∪ {𝑖})
5 df-un 3572 . . . . . 6 (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
64, 5eqtri 2642 . . . . 5 suc 𝑖 = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
7 df1o2 7557 . . . . . . 7 1𝑜 = {∅}
86, 7eleq12i 2692 . . . . . 6 (suc 𝑖 ∈ 1𝑜 ↔ {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅})
9 elsni 4185 . . . . . 6 ({𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
108, 9sylbi 207 . . . . 5 (suc 𝑖 ∈ 1𝑜 → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
116, 10syl5eq 2666 . . . 4 (suc 𝑖 ∈ 1𝑜 → suc 𝑖 = ∅)
123, 11mto 188 . . 3 ¬ suc 𝑖 ∈ 1𝑜
1312pm2.21i 116 . 2 (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
1413rgenw 2921 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 383   = wceq 1481  wcel 1988  {cab 2606  wral 2909  cun 3565  c0 3907  {csn 4168   ciun 4511  suc csuc 5713  cfv 5876  ωcom 7050  1𝑜c1o 7538   predc-bnj14 30728
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-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-v 3197  df-dif 3570  df-un 3572  df-nul 3908  df-sn 4169  df-suc 5717  df-1o 7545
This theorem is referenced by:  bnj150  30920
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