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Theorem infunsdom1 10190
Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 10191 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 769 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐴𝐵)
2 domsdomtr 9095 . . . . 5 ((𝐴𝐵𝐵 ≺ ω) → 𝐴 ≺ ω)
31, 2sylan 580 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
4 unfi2 9298 . . . 4 ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
53, 4sylancom 588 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
6 simpllr 774 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → ω ≼ 𝑋)
7 sdomdomtr 9093 . . 3 (((𝐴𝐵) ≺ ω ∧ ω ≼ 𝑋) → (𝐴𝐵) ≺ 𝑋)
85, 6, 7syl2anc 584 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ 𝑋)
9 omelon 9623 . . . . . 6 ω ∈ On
10 onenon 9926 . . . . . 6 (ω ∈ On → ω ∈ dom card)
119, 10ax-mp 5 . . . . 5 ω ∈ dom card
12 simpll 765 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝑋 ∈ dom card)
13 sdomdom 8959 . . . . . . 7 (𝐵𝑋𝐵𝑋)
1413ad2antll 727 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐵𝑋)
15 numdom 10015 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐵𝑋) → 𝐵 ∈ dom card)
1612, 14, 15syl2anc 584 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐵 ∈ dom card)
17 domtri2 9966 . . . . 5 ((ω ∈ dom card ∧ 𝐵 ∈ dom card) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
1811, 16, 17sylancr 587 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
1918biimpar 478 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ¬ 𝐵 ≺ ω) → ω ≼ 𝐵)
20 uncom 4149 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
2116adantr 481 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ∈ dom card)
22 simpr 485 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → ω ≼ 𝐵)
231adantr 481 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐴𝐵)
24 infunabs 10184 . . . . . 6 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2521, 22, 23, 24syl3anc 1371 . . . . 5 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐵𝐴) ≈ 𝐵)
2620, 25eqbrtrid 5176 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐴𝐵) ≈ 𝐵)
27 simplrr 776 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐵𝑋)
28 ensdomtr 9096 . . . 4 (((𝐴𝐵) ≈ 𝐵𝐵𝑋) → (𝐴𝐵) ≺ 𝑋)
2926, 27, 28syl2anc 584 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐴𝐵) ≺ 𝑋)
3019, 29syldan 591 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ¬ 𝐵 ≺ ω) → (𝐴𝐵) ≺ 𝑋)
318, 30pm2.61dan 811 1 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wcel 2106  cun 3942   class class class wbr 5141  dom cdm 5669  Oncon0 6353  ωcom 7838  cen 8919  cdom 8920  csdm 8921  cardccrd 9912
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-2o 8449  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-oi 9487  df-dju 9878  df-card 9916
This theorem is referenced by:  infunsdom  10191
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