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Theorem infunsdom1 10156
Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 10157 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 770 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐴𝐵)
2 domsdomtr 9063 . . . . 5 ((𝐴𝐵𝐵 ≺ ω) → 𝐴 ≺ ω)
31, 2sylan 581 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
4 unfi2 9266 . . . 4 ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
53, 4sylancom 589 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
6 simpllr 775 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → ω ≼ 𝑋)
7 sdomdomtr 9061 . . 3 (((𝐴𝐵) ≺ ω ∧ ω ≼ 𝑋) → (𝐴𝐵) ≺ 𝑋)
85, 6, 7syl2anc 585 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ 𝑋)
9 omelon 9589 . . . . . 6 ω ∈ On
10 onenon 9892 . . . . . 6 (ω ∈ On → ω ∈ dom card)
119, 10ax-mp 5 . . . . 5 ω ∈ dom card
12 simpll 766 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝑋 ∈ dom card)
13 sdomdom 8927 . . . . . . 7 (𝐵𝑋𝐵𝑋)
1413ad2antll 728 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐵𝑋)
15 numdom 9981 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐵𝑋) → 𝐵 ∈ dom card)
1612, 14, 15syl2anc 585 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐵 ∈ dom card)
17 domtri2 9932 . . . . 5 ((ω ∈ dom card ∧ 𝐵 ∈ dom card) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
1811, 16, 17sylancr 588 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
1918biimpar 479 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ¬ 𝐵 ≺ ω) → ω ≼ 𝐵)
20 uncom 4118 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
2116adantr 482 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ∈ dom card)
22 simpr 486 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → ω ≼ 𝐵)
231adantr 482 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐴𝐵)
24 infunabs 10150 . . . . . 6 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2521, 22, 23, 24syl3anc 1372 . . . . 5 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐵𝐴) ≈ 𝐵)
2620, 25eqbrtrid 5145 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐴𝐵) ≈ 𝐵)
27 simplrr 777 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐵𝑋)
28 ensdomtr 9064 . . . 4 (((𝐴𝐵) ≈ 𝐵𝐵𝑋) → (𝐴𝐵) ≺ 𝑋)
2926, 27, 28syl2anc 585 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐴𝐵) ≺ 𝑋)
3019, 29syldan 592 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ¬ 𝐵 ≺ ω) → (𝐴𝐵) ≺ 𝑋)
318, 30pm2.61dan 812 1 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wcel 2107  cun 3913   class class class wbr 5110  dom cdm 5638  Oncon0 6322  ωcom 7807  cen 8887  cdom 8888  csdm 8889  cardccrd 9878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9453  df-dju 9844  df-card 9882
This theorem is referenced by:  infunsdom  10157
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