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Theorem infunsdom1 9629
Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 9630 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 767 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐴𝐵)
2 domsdomtr 8646 . . . . 5 ((𝐴𝐵𝐵 ≺ ω) → 𝐴 ≺ ω)
31, 2sylan 580 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
4 unfi2 8781 . . . 4 ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
53, 4sylancom 588 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
6 simpllr 772 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → ω ≼ 𝑋)
7 sdomdomtr 8644 . . 3 (((𝐴𝐵) ≺ ω ∧ ω ≼ 𝑋) → (𝐴𝐵) ≺ 𝑋)
85, 6, 7syl2anc 584 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ 𝑋)
9 omelon 9103 . . . . . 6 ω ∈ On
10 onenon 9372 . . . . . 6 (ω ∈ On → ω ∈ dom card)
119, 10ax-mp 5 . . . . 5 ω ∈ dom card
12 simpll 763 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝑋 ∈ dom card)
13 sdomdom 8531 . . . . . . 7 (𝐵𝑋𝐵𝑋)
1413ad2antll 725 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐵𝑋)
15 numdom 9458 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐵𝑋) → 𝐵 ∈ dom card)
1612, 14, 15syl2anc 584 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐵 ∈ dom card)
17 domtri2 9412 . . . . 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 4133 . . . . 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 9623 . . . . . 6 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2521, 22, 23, 24syl3anc 1365 . . . . 5 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐵𝐴) ≈ 𝐵)
2620, 25eqbrtrid 5098 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐴𝐵) ≈ 𝐵)
27 simplrr 774 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐵𝑋)
28 ensdomtr 8647 . . . 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 809 1 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wcel 2107  cun 3938   class class class wbr 5063  dom cdm 5554  Oncon0 6190  ωcom 7573  cen 8500  cdom 8501  csdm 8502  cardccrd 9358
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-inf2 9098
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-oi 8968  df-dju 9324  df-card 9362
This theorem is referenced by:  infunsdom  9630
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