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Theorem onsetreclem3 48246
Description: Lemma for onsetrec 48247. (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 6375 . . . 4 (𝑎 ∈ On → Ord 𝑎)
2 orduniorsuc 7828 . . . 4 (Ord 𝑎 → (𝑎 = 𝑎𝑎 = suc 𝑎))
31, 2syl 17 . . 3 (𝑎 ∈ On → (𝑎 = 𝑎𝑎 = suc 𝑎))
4 vex 3467 . . . 4 𝑎 ∈ V
54elpr 4649 . . 3 (𝑎 ∈ { 𝑎, suc 𝑎} ↔ (𝑎 = 𝑎𝑎 = suc 𝑎))
63, 5sylibr 233 . 2 (𝑎 ∈ On → 𝑎 ∈ { 𝑎, suc 𝑎})
7 onsetreclem3.1 . . 3 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
87onsetreclem1 48244 . 2 (𝐹𝑎) = { 𝑎, suc 𝑎}
96, 8eleqtrrdi 2836 1 (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 845   = wceq 1533  wcel 2098  Vcvv 3463  {cpr 4627   cuni 4904  cmpt 5227  Ord word 6364  Oncon0 6365  suc csuc 6367  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  onsetrec  48247
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