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Theorem infunsdom 10149
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 8919 . . 3 (𝐴𝐵𝐴𝐵)
2 infunsdom1 10148 . . . . 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 593 . 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 8919 . . . . . . 7 (𝐵𝑋𝐵𝑋)
87ad2antll 727 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
9 numdom 9973 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐵𝑋) → 𝐵 ∈ dom card)
106, 8, 9syl2anc 584 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐵 ∈ dom card)
11 sdomdom 8919 . . . . . . 7 (𝐴𝑋𝐴𝑋)
1211ad2antrl 726 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
13 numdom 9973 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐴𝑋) → 𝐴 ∈ dom card)
146, 12, 13syl2anc 584 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐴 ∈ dom card)
15 domtri2 9924 . . . . 5 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1610, 14, 15syl2anc 584 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1716biimpar 478 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝐵) → 𝐵𝐴)
18 uncom 4113 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
19 infunsdom1 10148 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵𝐴𝐴𝑋)) → (𝐵𝐴) ≺ 𝑋)
2018, 19eqbrtrid 5140 . . . . 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 591 . 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 396  wcel 2106  cun 3908   class class class wbr 5105  dom cdm 5633  ωcom 7801  cdom 8880  csdm 8881  cardccrd 9870
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-inf2 9576
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7312  df-ov 7359  df-om 7802  df-1st 7920  df-2nd 7921  df-frecs 8211  df-wrecs 8242  df-recs 8316  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8647  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-oi 9445  df-dju 9836  df-card 9874
This theorem is referenced by:  csdfil  23243
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