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Theorem omabslem 7993
 Description: Lemma for omabs 7994. (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 7332 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
2 limom 7341 . . . . . . 7 Lim ω
32jctr 522 . . . . . 6 (ω ∈ On → (ω ∈ On ∧ Lim ω))
4 omlim 7880 . . . . . 6 ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·o ω) = 𝑥 ∈ ω (𝐴 ·o 𝑥))
51, 3, 4syl2an 591 . . . . 5 ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) = 𝑥 ∈ ω (𝐴 ·o 𝑥))
6 ordom 7335 . . . . . . . . 9 Ord ω
7 nnmcl 7959 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ∈ ω)
8 ordelss 5979 . . . . . . . . 9 ((Ord ω ∧ (𝐴 ·o 𝑥) ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
96, 7, 8sylancr 583 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
109ralrimiva 3175 . . . . . . 7 (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
11 iunss 4781 . . . . . . 7 ( 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
1210, 11sylibr 226 . . . . . 6 (𝐴 ∈ ω → 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
1312adantr 474 . . . . 5 ((𝐴 ∈ ω ∧ ω ∈ On) → 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
145, 13eqsstrd 3864 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) ⊆ ω)
1514ancoms 452 . . 3 ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 ·o ω) ⊆ ω)
16153adant3 1168 . 2 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) ⊆ ω)
17 omword2 7921 . . . 4 (((ω ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
18173impa 1142 . . 3 ((ω ∈ On ∧ 𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
191, 18syl3an2 1209 . 2 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
2016, 19eqssd 3844 1 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3117   ⊆ wss 3798  ∅c0 4144  ∪ ciun 4740  Ord word 5962  Oncon0 5963  Lim wlim 5964  (class class class)co 6905  ωcom 7326   ·o comu 7824 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-omul 7831 This theorem is referenced by:  omabs  7994
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