Theorem infunsdom1 10195

Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 10196 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 782	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐴 ≼ 𝐵)
2		domsdomtr 9100	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
3	1, 2	sylan 591	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
4		unfi2 9270	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
5	3, 4	sylancom 599	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
6		simpllr 787	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → ω ≼ 𝑋)
7		sdomdomtr 9098	. . 3 ⊢ (((𝐴 ∪ 𝐵) ≺ ω ∧ ω ≼ 𝑋) → (𝐴 ∪ 𝐵) ≺ 𝑋)
8	5, 6, 7	syl2anc 595	. 2 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ 𝑋)
9		omelon 9615	. . . . . 6 ⊢ ω ∈ On
10		onenon 9935	. . . . . 6 ⊢ (ω ∈ On → ω ∈ dom card)
11	9, 10	ax-mp 5	. . . . 5 ⊢ ω ∈ dom card
12		simpll 778	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝑋 ∈ dom card)
13		sdomdom 8977	. . . . . . 7 ⊢ (𝐵 ≺ 𝑋 → 𝐵 ≼ 𝑋)
14	13	ad2antll 741	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐵 ≼ 𝑋)
15		numdom 10022	. . . . . 6 ⊢ ((𝑋 ∈ dom card ∧ 𝐵 ≼ 𝑋) → 𝐵 ∈ dom card)
16	12, 14, 15	syl2anc 595	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐵 ∈ dom card)
17		domtri2 9975	. . . . 5 ⊢ ((ω ∈ dom card ∧ 𝐵 ∈ dom card) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
18	11, 16, 17	sylancr 598	. . . 4 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
19	18	biimpar 482	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ¬ 𝐵 ≺ ω) → ω ≼ 𝐵)
20		uncom 4120	. . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
21	16	adantr 485	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ∈ dom card)
22		simpr 489	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → ω ≼ 𝐵)
23	1	adantr 485	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐴 ≼ 𝐵)
24		infunabs 10189	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵 ∧ 𝐴 ≼ 𝐵) → (𝐵 ∪ 𝐴) ≈ 𝐵)
25	21, 22, 23, 24	syl3anc 1396	. . . . 5 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐵 ∪ 𝐴) ≈ 𝐵)
26	20, 25	eqbrtrid 5150	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐴 ∪ 𝐵) ≈ 𝐵)
27		simplrr 789	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ≺ 𝑋)
28		ensdomtr 9101	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ≈ 𝐵 ∧ 𝐵 ≺ 𝑋) → (𝐴 ∪ 𝐵) ≺ 𝑋)
29	26, 27, 28	syl2anc 595	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐴 ∪ 𝐵) ≺ 𝑋)
30	19, 29	syldan 602	. 2 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ¬ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ 𝑋)
31	8, 30	pm2.61dan 824	1 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2149   ∪ cun 3911   class class class wbr 5113  dom cdm 5662  Oncon0 6361  ωcom 7862   ≈ cen 8940   ≼ cdom 8941   ≺ csdm 8942  cardccrd 9921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-oi 9472  df-dju 9887  df-card 9925

This theorem is referenced by:  infunsdom  10196