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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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