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Theorem bnj98 33878
Description: Technical lemma for bnj150 33887. 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 3479 . . . . . 6 𝑖 ∈ V
21sucid 6447 . . . . 5 𝑖 ∈ suc 𝑖
32n0ii 4337 . . . 4 ¬ suc 𝑖 = ∅
4 df-suc 6371 . . . . . 6 suc 𝑖 = (𝑖 ∪ {𝑖})
5 df-un 3954 . . . . . 6 (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
64, 5eqtri 2761 . . . . 5 suc 𝑖 = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
7 df1o2 8473 . . . . . . 7 1o = {∅}
86, 7eleq12i 2827 . . . . . 6 (suc 𝑖 ∈ 1o ↔ {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅})
9 elsni 4646 . . . . . 6 ({𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
108, 9sylbi 216 . . . . 5 (suc 𝑖 ∈ 1o → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
116, 10eqtrid 2785 . . . 4 (suc 𝑖 ∈ 1o → suc 𝑖 = ∅)
123, 11mto 196 . . 3 ¬ suc 𝑖 ∈ 1o
1312pm2.21i 119 . 2 (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
1413rgenw 3066 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 846   = wceq 1542  wcel 2107  {cab 2710  wral 3062  cun 3947  c0 4323  {csn 4629   ciun 4998  suc csuc 6367  cfv 6544  ωcom 7855  1oc1o 8459   predc-bnj14 33699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-dif 3952  df-un 3954  df-nul 4324  df-sn 4630  df-suc 6371  df-1o 8466
This theorem is referenced by:  bnj150  33887
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