Theorem omabslem 8624

Description: Lemma for omabs 8625. (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 7856	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
2		limom 7866	. . . . . . 7 ⊢ Lim ω
3	2	jctr 533	. . . . . 6 ⊢ (ω ∈ On → (ω ∈ On ∧ Lim ω))
4		omlim 8506	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
5	1, 3, 4	syl2an 607	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
6		ordom 7860	. . . . . . . . 9 ⊢ Ord ω
7		nnmcl 8586	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ∈ ω)
8		ordelss 6365	. . . . . . . . 9 ⊢ ((Ord ω ∧ (𝐴 ·o 𝑥) ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
9	6, 7, 8	sylancr 598	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
10	9	ralrimiva 3157	. . . . . . 7 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
11		iunss 5004	. . . . . . 7 ⊢ (∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
12	10, 11	sylibr 237	. . . . . 6 ⊢ (𝐴 ∈ ω → ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
13	12	adantr 485	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
14	5, 13	eqsstrd 3973	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) ⊆ ω)
15	14	ancoms 463	. . 3 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 ·o ω) ⊆ ω)
16	15	3adant3 1148	. 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) ⊆ ω)
17		omword2 8547	. . . 4 ⊢ (((ω ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
18	17	3impa 1125	. . 3 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
19	1, 18	syl3an2 1180	. 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
20	16, 19	eqssd 3956	1 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079   ⊆ wss 3907  ∅c0 4288  ∪ ciun 4951  Ord word 6348  Oncon0 6349  Lim wlim 6350  (class class class)co 7400  ωcom 7850   ·_o comu 8439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-omul 8446

This theorem is referenced by:  omabs  8625  2omomeqom  43887  omnord1ex  43888