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Theorem bnj98 30645
 Description: Technical lemma for bnj150 30654. 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 3189 . . . . . 6 𝑖 ∈ V
21sucid 5763 . . . . 5 𝑖 ∈ suc 𝑖
32n0ii 3898 . . . 4 ¬ suc 𝑖 = ∅
4 df-suc 5688 . . . . . 6 suc 𝑖 = (𝑖 ∪ {𝑖})
5 df-un 3560 . . . . . 6 (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
64, 5eqtri 2643 . . . . 5 suc 𝑖 = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
7 df1o2 7517 . . . . . . 7 1𝑜 = {∅}
86, 7eleq12i 2691 . . . . . 6 (suc 𝑖 ∈ 1𝑜 ↔ {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅})
9 elsni 4165 . . . . . 6 ({𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
108, 9sylbi 207 . . . . 5 (suc 𝑖 ∈ 1𝑜 → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
116, 10syl5eq 2667 . . . 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 2919 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 1480   ∈ wcel 1987  {cab 2607  ∀wral 2907   ∪ cun 3553  ∅c0 3891  {csn 4148  ∪ ciun 4485  suc csuc 5684  ‘cfv 5847  ωcom 7012  1𝑜c1o 7498   predc-bnj14 30461 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-dif 3558  df-un 3560  df-nul 3892  df-sn 4149  df-suc 5688  df-1o 7505 This theorem is referenced by:  bnj150  30654
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