Theorem omabslem 8652

Description: Lemma for omabs 8653. (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.

Step	Hyp	Ref	Expression
1		nnon 7864	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
2		limom 7874	. . . . . . 7 ⊢ Lim ω
3	2	jctr 524	. . . . . 6 ⊢ (ω ∈ On → (ω ∈ On ∧ Lim ω))
4		omlim 8536	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
5	1, 3, 4	syl2an 595	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
6		ordom 7868	. . . . . . . . 9 ⊢ Ord ω
7		nnmcl 8615	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ∈ ω)
8		ordelss 6380	. . . . . . . . 9 ⊢ ((Ord ω ∧ (𝐴 ·o 𝑥) ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
9	6, 7, 8	sylancr 586	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
10	9	ralrimiva 3145	. . . . . . 7 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
11		iunss 5048	. . . . . . 7 ⊢ (∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
12	10, 11	sylibr 233	. . . . . 6 ⊢ (𝐴 ∈ ω → ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
13	12	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
14	5, 13	eqsstrd 4020	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) ⊆ ω)
15	14	ancoms 458	. . 3 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 ·o ω) ⊆ ω)
16	15	3adant3 1131	. 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) ⊆ ω)
17		omword2 8577	. . . 4 ⊢ (((ω ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
18	17	3impa 1109	. . 3 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
19	1, 18	syl3an2 1163	. 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
20	16, 19	eqssd 3999	1 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060   ⊆ wss 3948  ∅c0 4322  ∪ ciun 4997  Ord word 6363  Oncon0 6364  Lim wlim 6365  (class class class)co 7412  ωcom 7858   ·_o comu 8467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-oadd 8473  df-omul 8474

This theorem is referenced by:  omabs  8653  2omomeqom  42356  omnord1ex  42357