Theorem infunsdom1 10120

Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 10121 assumes additionally that 𝐴 is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Assertion

Ref	Expression
infunsdom1	⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)

Proof of Theorem infunsdom1

Step	Hyp	Ref	Expression
1		simprl 770	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐴 ≼ 𝐵)
2		domsdomtr 9038	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
3	1, 2	sylan 580	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
4		unfi2 9208	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
5	3, 4	sylancom 588	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
6		simpllr 775	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → ω ≼ 𝑋)
7		sdomdomtr 9036	. . 3 ⊢ (((𝐴 ∪ 𝐵) ≺ ω ∧ ω ≼ 𝑋) → (𝐴 ∪ 𝐵) ≺ 𝑋)
8	5, 6, 7	syl2anc 584	. 2 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ 𝑋)
9		omelon 9553	. . . . . 6 ⊢ ω ∈ On
10		onenon 9859	. . . . . 6 ⊢ (ω ∈ On → ω ∈ dom card)
11	9, 10	ax-mp 5	. . . . 5 ⊢ ω ∈ dom card
12		simpll 766	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝑋 ∈ dom card)
13		sdomdom 8915	. . . . . . 7 ⊢ (𝐵 ≺ 𝑋 → 𝐵 ≼ 𝑋)
14	13	ad2antll 729	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐵 ≼ 𝑋)
15		numdom 9946	. . . . . 6 ⊢ ((𝑋 ∈ dom card ∧ 𝐵 ≼ 𝑋) → 𝐵 ∈ dom card)
16	12, 14, 15	syl2anc 584	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐵 ∈ dom card)
17		domtri2 9899	. . . . 5 ⊢ ((ω ∈ dom card ∧ 𝐵 ∈ dom card) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
18	11, 16, 17	sylancr 587	. . . 4 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
19	18	biimpar 477	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ¬ 𝐵 ≺ ω) → ω ≼ 𝐵)
20		uncom 4108	. . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
21	16	adantr 480	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ∈ dom card)
22		simpr 484	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → ω ≼ 𝐵)
23	1	adantr 480	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐴 ≼ 𝐵)
24		infunabs 10114	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵 ∧ 𝐴 ≼ 𝐵) → (𝐵 ∪ 𝐴) ≈ 𝐵)
25	21, 22, 23, 24	syl3anc 1373	. . . . 5 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐵 ∪ 𝐴) ≈ 𝐵)
26	20, 25	eqbrtrid 5131	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐴 ∪ 𝐵) ≈ 𝐵)
27		simplrr 777	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ≺ 𝑋)
28		ensdomtr 9039	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ≈ 𝐵 ∧ 𝐵 ≺ 𝑋) → (𝐴 ∪ 𝐵) ≺ 𝑋)
29	26, 27, 28	syl2anc 584	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐴 ∪ 𝐵) ≺ 𝑋)
30	19, 29	syldan 591	. 2 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ¬ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ 𝑋)
31	8, 30	pm2.61dan 812	1 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113   ∪ cun 3897   class class class wbr 5096  dom cdm 5622  Oncon0 6315  ωcom 7806   ≈ cen 8878   ≼ cdom 8879   ≺ csdm 8880  cardccrd 9845

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 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548

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 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-oi 9413  df-dju 9811  df-card 9849

This theorem is referenced by:  infunsdom  10121