MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omordlim Structured version   Visualization version   GIF version

Theorem omordlim 8615
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 8571 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) → (𝐴 ·o 𝐵) = 𝑥𝐵 (𝐴 ·o 𝑥))
21eleq2d 2827 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) → (𝐶 ∈ (𝐴 ·o 𝐵) ↔ 𝐶 𝑥𝐵 (𝐴 ·o 𝑥)))
3 eliun 4995 . . 3 (𝐶 𝑥𝐵 (𝐴 ·o 𝑥) ↔ ∃𝑥𝐵 𝐶 ∈ (𝐴 ·o 𝑥))
42, 3bitrdi 287 . 2 ((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) → (𝐶 ∈ (𝐴 ·o 𝐵) ↔ ∃𝑥𝐵 𝐶 ∈ (𝐴 ·o 𝑥)))
54biimpa 476 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 395  wcel 2108  wrex 3070   ciun 4991  Oncon0 6384  Lim wlim 6385  (class class class)co 7431   ·o comu 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-omul 8511
This theorem is referenced by:  odi  8617  omass  8618  oaabs2  8687
  Copyright terms: Public domain W3C validator