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Theorem omabslem 8680
Description: Lemma for omabs 8681. (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 7882 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
2 limom 7892 . . . . . . 7 Lim ω
32jctr 523 . . . . . 6 (ω ∈ On → (ω ∈ On ∧ Lim ω))
4 omlim 8563 . . . . . 6 ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·o ω) = 𝑥 ∈ ω (𝐴 ·o 𝑥))
51, 3, 4syl2an 594 . . . . 5 ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) = 𝑥 ∈ ω (𝐴 ·o 𝑥))
6 ordom 7886 . . . . . . . . 9 Ord ω
7 nnmcl 8642 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ∈ ω)
8 ordelss 6392 . . . . . . . . 9 ((Ord ω ∧ (𝐴 ·o 𝑥) ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
96, 7, 8sylancr 585 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
109ralrimiva 3136 . . . . . . 7 (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
11 iunss 5053 . . . . . . 7 ( 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
1210, 11sylibr 233 . . . . . 6 (𝐴 ∈ ω → 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
1312adantr 479 . . . . 5 ((𝐴 ∈ ω ∧ ω ∈ On) → 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
145, 13eqsstrd 4018 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) ⊆ ω)
1514ancoms 457 . . 3 ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 ·o ω) ⊆ ω)
16153adant3 1129 . 2 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) ⊆ ω)
17 omword2 8604 . . . 4 (((ω ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
18173impa 1107 . . 3 ((ω ∈ On ∧ 𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
191, 18syl3an2 1161 . 2 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
2016, 19eqssd 3997 1 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wss 3947  c0 4325   ciun 5001  Ord word 6375  Oncon0 6376  Lim wlim 6377  (class class class)co 7424  ωcom 7876   ·o comu 8494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-oadd 8500  df-omul 8501
This theorem is referenced by:  omabs  8681  2omomeqom  42969  omnord1ex  42970
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