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Theorem bnj98 31275
 Description: Technical lemma for bnj150 31284. 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 3354 . . . . . 6 𝑖 ∈ V
21sucid 5947 . . . . 5 𝑖 ∈ suc 𝑖
32n0ii 4070 . . . 4 ¬ suc 𝑖 = ∅
4 df-suc 5872 . . . . . 6 suc 𝑖 = (𝑖 ∪ {𝑖})
5 df-un 3728 . . . . . 6 (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
64, 5eqtri 2793 . . . . 5 suc 𝑖 = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
7 df1o2 7726 . . . . . . 7 1𝑜 = {∅}
86, 7eleq12i 2843 . . . . . 6 (suc 𝑖 ∈ 1𝑜 ↔ {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅})
9 elsni 4333 . . . . . 6 ({𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
108, 9sylbi 207 . . . . 5 (suc 𝑖 ∈ 1𝑜 → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
116, 10syl5eq 2817 . . . 4 (suc 𝑖 ∈ 1𝑜 → suc 𝑖 = ∅)
123, 11mto 188 . . 3 ¬ suc 𝑖 ∈ 1𝑜
1312pm2.21i 117 . 2 (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
1413rgenw 3073 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 834   = wceq 1631   ∈ wcel 2145  {cab 2757  ∀wral 3061   ∪ cun 3721  ∅c0 4063  {csn 4316  ∪ ciun 4654  suc csuc 5868  ‘cfv 6031  ωcom 7212  1𝑜c1o 7706   predc-bnj14 31094 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-sn 4317  df-suc 5872  df-1o 7713 This theorem is referenced by:  bnj150  31284
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