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Theorem omordlim 7602
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 7558 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥))
21eleq2d 2684 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) → (𝐶 ∈ (𝐴 ·𝑜 𝐵) ↔ 𝐶 𝑥𝐵 (𝐴 ·𝑜 𝑥)))
3 eliun 4490 . . 3 (𝐶 𝑥𝐵 (𝐴 ·𝑜 𝑥) ↔ ∃𝑥𝐵 𝐶 ∈ (𝐴 ·𝑜 𝑥))
42, 3syl6bb 276 . 2 ((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) → (𝐶 ∈ (𝐴 ·𝑜 𝐵) ↔ ∃𝑥𝐵 𝐶 ∈ (𝐴 ·𝑜 𝑥)))
54biimpa 501 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 384  wcel 1987  wrex 2908   ciun 4485  Oncon0 5682  Lim wlim 5683  (class class class)co 6604   ·𝑜 comu 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-omul 7510
This theorem is referenced by:  odi  7604  omass  7605  oaabs2  7670
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