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Theorem infunsdom 8988
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) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)

Proof of Theorem infunsdom
StepHypRef Expression
1 sdomdom 7935 . . 3 (𝐴𝐵𝐴𝐵)
2 infunsdom1 8987 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
32anass1rs 848 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝐵𝑋) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
43adantlrl 755 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
51, 4sylan2 491 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
6 simpll 789 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝑋 ∈ dom card)
7 sdomdom 7935 . . . . . . 7 (𝐵𝑋𝐵𝑋)
87ad2antll 764 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
9 numdom 8813 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐵𝑋) → 𝐵 ∈ dom card)
106, 8, 9syl2anc 692 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐵 ∈ dom card)
11 sdomdom 7935 . . . . . . 7 (𝐴𝑋𝐴𝑋)
1211ad2antrl 763 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
13 numdom 8813 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐴𝑋) → 𝐴 ∈ dom card)
146, 12, 13syl2anc 692 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐴 ∈ dom card)
15 domtri2 8767 . . . . 5 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1610, 14, 15syl2anc 692 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1716biimpar 502 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝐵) → 𝐵𝐴)
18 uncom 3740 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
19 infunsdom1 8987 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵𝐴𝐴𝑋)) → (𝐵𝐴) ≺ 𝑋)
2018, 19syl5eqbr 4653 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵𝐴𝐴𝑋)) → (𝐴𝐵) ≺ 𝑋)
2120anass1rs 848 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝐴𝑋) ∧ 𝐵𝐴) → (𝐴𝐵) ≺ 𝑋)
2221adantlrr 756 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐵𝐴) → (𝐴𝐵) ≺ 𝑋)
2317, 22syldan 487 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
245, 23pm2.61dan 831 1 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wcel 1987  cun 3557   class class class wbr 4618  dom cdm 5079  ωcom 7019  cdom 7905  csdm 7906  cardccrd 8713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-oi 8367  df-card 8717  df-cda 8942
This theorem is referenced by:  csdfil  21621
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