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Theorem infunsdom1 9827
Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 9828 assumes additionally that 𝐴 is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
infunsdom1 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)

Proof of Theorem infunsdom1
StepHypRef Expression
1 simprl 771 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐴𝐵)
2 domsdomtr 8781 . . . . 5 ((𝐴𝐵𝐵 ≺ ω) → 𝐴 ≺ ω)
31, 2sylan 583 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
4 unfi2 8940 . . . 4 ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
53, 4sylancom 591 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
6 simpllr 776 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → ω ≼ 𝑋)
7 sdomdomtr 8779 . . 3 (((𝐴𝐵) ≺ ω ∧ ω ≼ 𝑋) → (𝐴𝐵) ≺ 𝑋)
85, 6, 7syl2anc 587 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ 𝑋)
9 omelon 9261 . . . . . 6 ω ∈ On
10 onenon 9565 . . . . . 6 (ω ∈ On → ω ∈ dom card)
119, 10ax-mp 5 . . . . 5 ω ∈ dom card
12 simpll 767 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝑋 ∈ dom card)
13 sdomdom 8656 . . . . . . 7 (𝐵𝑋𝐵𝑋)
1413ad2antll 729 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐵𝑋)
15 numdom 9652 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐵𝑋) → 𝐵 ∈ dom card)
1612, 14, 15syl2anc 587 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐵 ∈ dom card)
17 domtri2 9605 . . . . 5 ((ω ∈ dom card ∧ 𝐵 ∈ dom card) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
1811, 16, 17sylancr 590 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
1918biimpar 481 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ¬ 𝐵 ≺ ω) → ω ≼ 𝐵)
20 uncom 4067 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
2116adantr 484 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ∈ dom card)
22 simpr 488 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → ω ≼ 𝐵)
231adantr 484 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐴𝐵)
24 infunabs 9821 . . . . . 6 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2521, 22, 23, 24syl3anc 1373 . . . . 5 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐵𝐴) ≈ 𝐵)
2620, 25eqbrtrid 5088 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐴𝐵) ≈ 𝐵)
27 simplrr 778 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐵𝑋)
28 ensdomtr 8782 . . . 4 (((𝐴𝐵) ≈ 𝐵𝐵𝑋) → (𝐴𝐵) ≺ 𝑋)
2926, 27, 28syl2anc 587 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐴𝐵) ≺ 𝑋)
3019, 29syldan 594 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ¬ 𝐵 ≺ ω) → (𝐴𝐵) ≺ 𝑋)
318, 30pm2.61dan 813 1 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2110  cun 3864   class class class wbr 5053  dom cdm 5551  Oncon0 6213  ωcom 7644  cen 8623  cdom 8624  csdm 8625  cardccrd 9551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-oi 9126  df-dju 9517  df-card 9555
This theorem is referenced by:  infunsdom  9828
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