Theorem omabslem 8586

Description: Lemma for omabs 8587. (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 7823	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
2		limom 7833	. . . . . . 7 ⊢ Lim ω
3	2	jctr 524	. . . . . 6 ⊢ (ω ∈ On → (ω ∈ On ∧ Lim ω))
4		omlim 8468	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
5	1, 3, 4	syl2an 597	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
6		ordom 7827	. . . . . . . . 9 ⊢ Ord ω
7		nnmcl 8548	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ∈ ω)
8		ordelss 6339	. . . . . . . . 9 ⊢ ((Ord ω ∧ (𝐴 ·o 𝑥) ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
9	6, 7, 8	sylancr 588	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
10	9	ralrimiva 3129	. . . . . . 7 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
11		iunss 4987	. . . . . . 7 ⊢ (∪ 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
12	10, 11	sylibr 234	. . . . . 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 3956	. . . 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 1133	. 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) ⊆ ω)
17		omword2 8509	. . . 4 ⊢ (((ω ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
18	17	3impa 1110	. . 3 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
19	1, 18	syl3an2 1165	. 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
20	16, 19	eqssd 3939	1 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  ∅c0 4273  ∪ ciun 4933  Ord word 6322  Oncon0 6323  Lim wlim 6324  (class class class)co 7367  ωcom 7817   ·_o comu 8403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410

This theorem is referenced by:  omabs  8587  2omomeqom  43731  omnord1ex  43732