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Theorem bnj98 31260
Description: Technical lemma for bnj150 31269. 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 3394 . . . . . 6 𝑖 ∈ V
21sucid 6017 . . . . 5 𝑖 ∈ suc 𝑖
32n0ii 4124 . . . 4 ¬ suc 𝑖 = ∅
4 df-suc 5942 . . . . . 6 suc 𝑖 = (𝑖 ∪ {𝑖})
5 df-un 3774 . . . . . 6 (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
64, 5eqtri 2828 . . . . 5 suc 𝑖 = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
7 df1o2 7809 . . . . . . 7 1𝑜 = {∅}
86, 7eleq12i 2878 . . . . . 6 (suc 𝑖 ∈ 1𝑜 ↔ {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅})
9 elsni 4387 . . . . . 6 ({𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
108, 9sylbi 208 . . . . 5 (suc 𝑖 ∈ 1𝑜 → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
116, 10syl5eq 2852 . . . 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 3112 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 865   = wceq 1637  wcel 2156  {cab 2792  wral 3096  cun 3767  c0 4116  {csn 4370   ciun 4712  suc csuc 5938  cfv 6101  ωcom 7295  1𝑜c1o 7789   predc-bnj14 31079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-v 3393  df-dif 3772  df-un 3774  df-nul 4117  df-sn 4371  df-suc 5942  df-1o 7796
This theorem is referenced by:  bnj150  31269
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