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Theorem omordlim 8199
Description: Ordering involving the product of a limit ordinal. Proposition 8.23 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)
Assertion
Ref Expression
omordlim (((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) ∧ 𝐶 ∈ (𝐴 ·o 𝐵)) → ∃𝑥𝐵 𝐶 ∈ (𝐴 ·o 𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶
Allowed substitution hint:   𝐷(𝑥)

Proof of Theorem omordlim
StepHypRef Expression
1 omlim 8154 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) → (𝐴 ·o 𝐵) = 𝑥𝐵 (𝐴 ·o 𝑥))
21eleq2d 2901 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) → (𝐶 ∈ (𝐴 ·o 𝐵) ↔ 𝐶 𝑥𝐵 (𝐴 ·o 𝑥)))
3 eliun 4909 . . 3 (𝐶 𝑥𝐵 (𝐴 ·o 𝑥) ↔ ∃𝑥𝐵 𝐶 ∈ (𝐴 ·o 𝑥))
42, 3syl6bb 290 . 2 ((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) → (𝐶 ∈ (𝐴 ·o 𝐵) ↔ ∃𝑥𝐵 𝐶 ∈ (𝐴 ·o 𝑥)))
54biimpa 480 1 (((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) ∧ 𝐶 ∈ (𝐴 ·o 𝐵)) → ∃𝑥𝐵 𝐶 ∈ (𝐴 ·o 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wrex 3134   ciun 4905  Oncon0 6178  Lim wlim 6179  (class class class)co 7149   ·o comu 8096
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-omul 8103
This theorem is referenced by:  odi  8201  omass  8202  oaabs2  8268
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