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Theorem onsetreclem3 45277
 Description: Lemma for onsetrec 45278. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
Hypothesis
Ref Expression
onsetreclem3.1 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
Assertion
Ref Expression
onsetreclem3 (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
Distinct variable group:   𝑥,𝑎
Allowed substitution hints:   𝐹(𝑥,𝑎)

Proof of Theorem onsetreclem3
StepHypRef Expression
1 eloni 6170 . . . 4 (𝑎 ∈ On → Ord 𝑎)
2 orduniorsuc 7528 . . . 4 (Ord 𝑎 → (𝑎 = 𝑎𝑎 = suc 𝑎))
31, 2syl 17 . . 3 (𝑎 ∈ On → (𝑎 = 𝑎𝑎 = suc 𝑎))
4 vex 3444 . . . 4 𝑎 ∈ V
54elpr 4548 . . 3 (𝑎 ∈ { 𝑎, suc 𝑎} ↔ (𝑎 = 𝑎𝑎 = suc 𝑎))
63, 5sylibr 237 . 2 (𝑎 ∈ On → 𝑎 ∈ { 𝑎, suc 𝑎})
7 onsetreclem3.1 . . 3 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
87onsetreclem1 45275 . 2 (𝐹𝑎) = { 𝑎, suc 𝑎}
96, 8eleqtrrdi 2901 1 (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {cpr 4527  ∪ cuni 4801   ↦ cmpt 5111  Ord word 6159  Oncon0 6160  suc csuc 6162  ‘cfv 6325 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-ord 6163  df-on 6164  df-suc 6166  df-iota 6284  df-fun 6327  df-fv 6333 This theorem is referenced by:  onsetrec  45278
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