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Theorem infunsdom 8919
 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 7869 . . 3 (𝐴𝐵𝐴𝐵)
2 infunsdom1 8918 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
32anass1rs 845 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝐵𝑋) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
43adantlrl 752 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
51, 4sylan2 490 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
6 simpll 786 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝑋 ∈ dom card)
7 sdomdom 7869 . . . . . . 7 (𝐵𝑋𝐵𝑋)
87ad2antll 761 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
9 numdom 8744 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐵𝑋) → 𝐵 ∈ dom card)
106, 8, 9syl2anc 691 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐵 ∈ dom card)
11 sdomdom 7869 . . . . . . 7 (𝐴𝑋𝐴𝑋)
1211ad2antrl 760 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
13 numdom 8744 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐴𝑋) → 𝐴 ∈ dom card)
146, 12, 13syl2anc 691 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐴 ∈ dom card)
15 domtri2 8698 . . . . 5 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1610, 14, 15syl2anc 691 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1716biimpar 501 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝐵) → 𝐵𝐴)
18 uncom 3719 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
19 infunsdom1 8918 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵𝐴𝐴𝑋)) → (𝐵𝐴) ≺ 𝑋)
2018, 19syl5eqbr 4618 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵𝐴𝐴𝑋)) → (𝐴𝐵) ≺ 𝑋)
2120anass1rs 845 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝐴𝑋) ∧ 𝐵𝐴) → (𝐴𝐵) ≺ 𝑋)
2221adantlrr 753 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐵𝐴) → (𝐴𝐵) ≺ 𝑋)
2317, 22syldan 486 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
245, 23pm2.61dan 828 1 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ∪ cun 3538   class class class wbr 4583  dom cdm 5038  ωcom 6957   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-cda 8873 This theorem is referenced by:  csdfil  21508
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