Theorem omabslem 8571

Description: Lemma for omabs 8572. (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 7808	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
2		limom 7818	. . . . . . 7 ⊢ Lim ω
3	2	jctr 524	. . . . . 6 ⊢ (ω ∈ On → (ω ∈ On ∧ Lim ω))
4		omlim 8454	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
5	1, 3, 4	syl2an 596	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 ·o ω) = ∪ 𝑥 ∈ ω (𝐴 ·o 𝑥))
6		ordom 7812	. . . . . . . . 9 ⊢ Ord ω
7		nnmcl 8533	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ∈ ω)
8		ordelss 6327	. . . . . . . . 9 ⊢ ((Ord ω ∧ (𝐴 ·o 𝑥) ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
9	6, 7, 8	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ·o 𝑥) ⊆ ω)
10	9	ralrimiva 3125	. . . . . . 7 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ ω)
11		iunss 4995	. . . . . . 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 3965	. . . 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 1132	. 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) ⊆ ω)
17		omword2 8495	. . . 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 1164	. 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → ω ⊆ (𝐴 ·o ω))
20	16, 19	eqssd 3948	1 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ⊆ wss 3898  ∅c0 4282  ∪ ciun 4941  Ord word 6310  Oncon0 6311  Lim wlim 6312  (class class class)co 7352  ωcom 7802   ·_o comu 8389

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-oadd 8395  df-omul 8396

This theorem is referenced by:  omabs  8572  2omomeqom  43421  omnord1ex  43422