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Theorem omabslem 8265
Description: Lemma for omabs 8266. (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 7578 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
2 limom 7587 . . . . . . 7 Lim ω
32jctr 527 . . . . . 6 (ω ∈ On → (ω ∈ On ∧ Lim ω))
4 omlim 8150 . . . . . 6 ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·o ω) = 𝑥 ∈ ω (𝐴 ·o 𝑥))
51, 3, 4syl2an 597 . . . . 5 ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) = 𝑥 ∈ ω (𝐴 ·o 𝑥))
6 ordom 7581 . . . . . . . . 9 Ord ω
7 nnmcl 8230 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ∈ ω)
8 ordelss 6200 . . . . . . . . 9 ((Ord ω ∧ (𝐴 ·o 𝑥) ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
96, 7, 8sylancr 589 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
109ralrimiva 3180 . . . . . . 7 (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
11 iunss 4960 . . . . . . 7 ( 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
1210, 11sylibr 236 . . . . . 6 (𝐴 ∈ ω → 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
1312adantr 483 . . . . 5 ((𝐴 ∈ ω ∧ ω ∈ On) → 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
145, 13eqsstrd 4003 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) ⊆ ω)
1514ancoms 461 . . 3 ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 ·o ω) ⊆ ω)
16153adant3 1126 . 2 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) ⊆ ω)
17 omword2 8192 . . . 4 (((ω ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
18173impa 1104 . . 3 ((ω ∈ On ∧ 𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
191, 18syl3an2 1158 . 2 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
2016, 19eqssd 3982 1 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  wss 3934  c0 4289   ciun 4910  Ord word 6183  Oncon0 6184  Lim wlim 6185  (class class class)co 7148  ωcom 7572   ·o comu 8092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-omul 8099
This theorem is referenced by:  omabs  8266
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