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Theorem bnj98 31265
Description: Technical lemma for bnj150 31274. 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 3343 . . . . . 6 𝑖 ∈ V
21sucid 5965 . . . . 5 𝑖 ∈ suc 𝑖
32n0ii 4065 . . . 4 ¬ suc 𝑖 = ∅
4 df-suc 5890 . . . . . 6 suc 𝑖 = (𝑖 ∪ {𝑖})
5 df-un 3720 . . . . . 6 (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
64, 5eqtri 2782 . . . . 5 suc 𝑖 = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
7 df1o2 7743 . . . . . . 7 1𝑜 = {∅}
86, 7eleq12i 2832 . . . . . 6 (suc 𝑖 ∈ 1𝑜 ↔ {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅})
9 elsni 4338 . . . . . 6 ({𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
108, 9sylbi 207 . . . . 5 (suc 𝑖 ∈ 1𝑜 → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
116, 10syl5eq 2806 . . . 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 3062 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 382   = wceq 1632  wcel 2139  {cab 2746  wral 3050  cun 3713  c0 4058  {csn 4321   ciun 4672  suc csuc 5886  cfv 6049  ωcom 7231  1𝑜c1o 7723   predc-bnj14 31084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-suc 5890  df-1o 7730
This theorem is referenced by:  bnj150  31274
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