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Theorem infunsdom 10237
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 8999 . . 3 (𝐴𝐵𝐴𝐵)
2 infunsdom1 10236 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
32anass1rs 653 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝐵𝑋) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
43adantlrl 718 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
51, 4sylan2 591 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
6 simpll 765 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝑋 ∈ dom card)
7 sdomdom 8999 . . . . . . 7 (𝐵𝑋𝐵𝑋)
87ad2antll 727 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
9 numdom 10061 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐵𝑋) → 𝐵 ∈ dom card)
106, 8, 9syl2anc 582 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐵 ∈ dom card)
11 sdomdom 8999 . . . . . . 7 (𝐴𝑋𝐴𝑋)
1211ad2antrl 726 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
13 numdom 10061 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐴𝑋) → 𝐴 ∈ dom card)
146, 12, 13syl2anc 582 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐴 ∈ dom card)
15 domtri2 10012 . . . . 5 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1610, 14, 15syl2anc 582 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1716biimpar 476 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝐵) → 𝐵𝐴)
18 uncom 4146 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
19 infunsdom1 10236 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵𝐴𝐴𝑋)) → (𝐵𝐴) ≺ 𝑋)
2018, 19eqbrtrid 5178 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵𝐴𝐴𝑋)) → (𝐴𝐵) ≺ 𝑋)
2120anass1rs 653 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝐴𝑋) ∧ 𝐵𝐴) → (𝐴𝐵) ≺ 𝑋)
2221adantlrr 719 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐵𝐴) → (𝐴𝐵) ≺ 𝑋)
2317, 22syldan 589 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
245, 23pm2.61dan 811 1 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wcel 2098  cun 3937   class class class wbr 5143  dom cdm 5672  ωcom 7868  cdom 8960  csdm 8961  cardccrd 9958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-oi 9533  df-dju 9924  df-card 9962
This theorem is referenced by:  csdfil  23816
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