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Theorem onsetreclem2 42248
 Description: Lemma for onsetrec 42250. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
Hypothesis
Ref Expression
onsetreclem2.1 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
Assertion
Ref Expression
onsetreclem2 (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
Distinct variable group:   𝑥,𝑎
Allowed substitution hints:   𝐹(𝑥,𝑎)

Proof of Theorem onsetreclem2
StepHypRef Expression
1 onsetreclem2.1 . . 3 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
21onsetreclem1 42247 . 2 (𝐹𝑎) = { 𝑎, suc 𝑎}
3 vex 3176 . . . 4 𝑎 ∈ V
43ssonunii 6879 . . 3 (𝑎 ⊆ On → 𝑎 ∈ On)
5 suceloni 6905 . . . 4 ( 𝑎 ∈ On → suc 𝑎 ∈ On)
6 prssi 4293 . . . 4 (( 𝑎 ∈ On ∧ suc 𝑎 ∈ On) → { 𝑎, suc 𝑎} ⊆ On)
75, 6mpdan 699 . . 3 ( 𝑎 ∈ On → { 𝑎, suc 𝑎} ⊆ On)
84, 7syl 17 . 2 (𝑎 ⊆ On → { 𝑎, suc 𝑎} ⊆ On)
92, 8syl5eqss 3612 1 (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  {cpr 4127  ∪ cuni 4372   ↦ cmpt 4643  Oncon0 5640  suc csuc 5642  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  onsetrec  42250
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