Theorem infunsdom 9674

Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also strictly dominated by 𝑋. (Contributed by Mario Carneiro, 3-May-2015.)

Assertion

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

Proof of Theorem infunsdom

Step	Hyp	Ref	Expression
1		sdomdom 8555	. . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵)
2		infunsdom1 9673	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)
3	2	anass1rs 654	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝐵 ≺ 𝑋) ∧ 𝐴 ≼ 𝐵) → (𝐴 ∪ 𝐵) ≺ 𝑋)
4	3	adantlrl 719	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐴 ≼ 𝐵) → (𝐴 ∪ 𝐵) ≺ 𝑋)
5	1, 4	sylan2 595	. 2 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐴 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≺ 𝑋)
6		simpll 766	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) → 𝑋 ∈ dom card)
7		sdomdom 8555	. . . . . . 7 ⊢ (𝐵 ≺ 𝑋 → 𝐵 ≼ 𝑋)
8	7	ad2antll 728	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) → 𝐵 ≼ 𝑋)
9		numdom 9498	. . . . . 6 ⊢ ((𝑋 ∈ dom card ∧ 𝐵 ≼ 𝑋) → 𝐵 ∈ dom card)
10	6, 8, 9	syl2anc 587	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) → 𝐵 ∈ dom card)
11		sdomdom 8555	. . . . . . 7 ⊢ (𝐴 ≺ 𝑋 → 𝐴 ≼ 𝑋)
12	11	ad2antrl 727	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) → 𝐴 ≼ 𝑋)
13		numdom 9498	. . . . . 6 ⊢ ((𝑋 ∈ dom card ∧ 𝐴 ≼ 𝑋) → 𝐴 ∈ dom card)
14	6, 12, 13	syl2anc 587	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) → 𝐴 ∈ dom card)
15		domtri2 9451	. . . . 5 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → (𝐵 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝐵))
16	10, 14, 15	syl2anc 587	. . . 4 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) → (𝐵 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝐵))
17	16	biimpar 481	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) ∧ ¬ 𝐴 ≺ 𝐵) → 𝐵 ≼ 𝐴)
18		uncom 4058	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
19		infunsdom1 9673	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵 ≼ 𝐴 ∧ 𝐴 ≺ 𝑋)) → (𝐵 ∪ 𝐴) ≺ 𝑋)
20	18, 19	eqbrtrid 5067	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵 ≼ 𝐴 ∧ 𝐴 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)
21	20	anass1rs 654	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝐴 ≺ 𝑋) ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≺ 𝑋)
22	21	adantlrr 720	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≺ 𝑋)
23	17, 22	syldan 594	. 2 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) ∧ ¬ 𝐴 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≺ 𝑋)
24	5, 23	pm2.61dan 812	1 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111   ∪ cun 3856   class class class wbr 5032  dom cdm 5524  ωcom 7579   ≼ cdom 8525   ≺ csdm 8526  cardccrd 9397

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-inf2 9137

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-oi 9007  df-dju 9363  df-card 9401

This theorem is referenced by:  csdfil  22594