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Theorem omabslem 7686
Description: Lemma for omabs 7687. (Contributed by Mario Carneiro, 30-May-2015.)
Assertion
Ref Expression
omabslem ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 ω) = ω)

Proof of Theorem omabslem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nnon 7033 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
2 limom 7042 . . . . . . 7 Lim ω
32jctr 564 . . . . . 6 (ω ∈ On → (ω ∈ On ∧ Lim ω))
4 omlim 7573 . . . . . 6 ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·𝑜 ω) = 𝑥 ∈ ω (𝐴 ·𝑜 𝑥))
51, 3, 4syl2an 494 . . . . 5 ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·𝑜 ω) = 𝑥 ∈ ω (𝐴 ·𝑜 𝑥))
6 ordom 7036 . . . . . . . . 9 Ord ω
7 nnmcl 7652 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·𝑜 𝑥) ∈ ω)
8 ordelss 5708 . . . . . . . . 9 ((Ord ω ∧ (𝐴 ·𝑜 𝑥) ∈ ω) → (𝐴 ·𝑜 𝑥) ⊆ ω)
96, 7, 8sylancr 694 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·𝑜 𝑥) ⊆ ω)
109ralrimiva 2962 . . . . . . 7 (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 ·𝑜 𝑥) ⊆ ω)
11 iunss 4534 . . . . . . 7 ( 𝑥 ∈ ω (𝐴 ·𝑜 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 ·𝑜 𝑥) ⊆ ω)
1210, 11sylibr 224 . . . . . 6 (𝐴 ∈ ω → 𝑥 ∈ ω (𝐴 ·𝑜 𝑥) ⊆ ω)
1312adantr 481 . . . . 5 ((𝐴 ∈ ω ∧ ω ∈ On) → 𝑥 ∈ ω (𝐴 ·𝑜 𝑥) ⊆ ω)
145, 13eqsstrd 3624 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·𝑜 ω) ⊆ ω)
1514ancoms 469 . . 3 ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 ·𝑜 ω) ⊆ ω)
16153adant3 1079 . 2 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 ω) ⊆ ω)
17 omword2 7614 . . . 4 (((ω ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·𝑜 ω))
18173impa 1256 . . 3 ((ω ∈ On ∧ 𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·𝑜 ω))
191, 18syl3an2 1357 . 2 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·𝑜 ω))
2016, 19eqssd 3605 1 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 ω) = ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wss 3560  c0 3897   ciun 4492  Ord word 5691  Oncon0 5692  Lim wlim 5693  (class class class)co 6615  ωcom 7027   ·𝑜 comu 7518
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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525
This theorem is referenced by:  omabs  7687
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