Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj98 Structured version   Visualization version   GIF version

Theorem bnj98 31751
 Description: Technical lemma for bnj150 31760. 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 3443 . . . . . 6 𝑖 ∈ V
21sucid 6152 . . . . 5 𝑖 ∈ suc 𝑖
32n0ii 4228 . . . 4 ¬ suc 𝑖 = ∅
4 df-suc 6079 . . . . . 6 suc 𝑖 = (𝑖 ∪ {𝑖})
5 df-un 3870 . . . . . 6 (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
64, 5eqtri 2821 . . . . 5 suc 𝑖 = {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})}
7 df1o2 7974 . . . . . . 7 1o = {∅}
86, 7eleq12i 2877 . . . . . 6 (suc 𝑖 ∈ 1o ↔ {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅})
9 elsni 4495 . . . . . 6 ({𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
108, 9sylbi 218 . . . . 5 (suc 𝑖 ∈ 1o → {𝑥 ∣ (𝑥𝑖𝑥 ∈ {𝑖})} = ∅)
116, 10syl5eq 2845 . . . 4 (suc 𝑖 ∈ 1o → suc 𝑖 = ∅)
123, 11mto 198 . . 3 ¬ suc 𝑖 ∈ 1o
1312pm2.21i 119 . 2 (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
1413rgenw 3119 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 842   = wceq 1525   ∈ wcel 2083  {cab 2777  ∀wral 3107   ∪ cun 3863  ∅c0 4217  {csn 4478  ∪ ciun 4831  suc csuc 6075  ‘cfv 6232  ωcom 7443  1oc1o 7953   predc-bnj14 31571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-v 3442  df-dif 3868  df-un 3870  df-nul 4218  df-sn 4479  df-suc 6079  df-1o 7960 This theorem is referenced by:  bnj150  31760
 Copyright terms: Public domain W3C validator