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Theorem rp-oelim2 43298
Description: The power of an ordinal at least as large as two with a limit ordinal on thr right is a limit ordinal. Lemma 3.21 of [Schloeder] p. 10. See oelimcl 8637. (Contributed by RP, 30-Jan-2025.)
Assertion
Ref Expression
rp-oelim2 (((𝐴 ∈ On ∧ 1o𝐴) ∧ (Lim 𝐵𝐵𝑉)) → Lim (𝐴o 𝐵))

Proof of Theorem rp-oelim2
StepHypRef Expression
1 ondif2 8539 . . 3 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
21biimpri 228 . 2 ((𝐴 ∈ On ∧ 1o𝐴) → 𝐴 ∈ (On ∖ 2o))
3 pm3.22 459 . 2 ((Lim 𝐵𝐵𝑉) → (𝐵𝑉 ∧ Lim 𝐵))
4 oelimcl 8637 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝑉 ∧ Lim 𝐵)) → Lim (𝐴o 𝐵))
52, 3, 4syl2an 596 1 (((𝐴 ∈ On ∧ 1o𝐴) ∧ (Lim 𝐵𝐵𝑉)) → Lim (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  cdif 3960  Oncon0 6386  Lim wlim 6387  (class class class)co 7431  1oc1o 8498  2oc2o 8499  o coe 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511
This theorem is referenced by: (None)
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