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Theorem infunsdom1 9258
Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 9259 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 776 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐴𝐵)
2 domsdomtr 8272 . . . . 5 ((𝐴𝐵𝐵 ≺ ω) → 𝐴 ≺ ω)
31, 2sylan 570 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
4 unfi2 8406 . . . 4 ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
53, 4sylancom 577 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)
6 simpllr 782 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → ω ≼ 𝑋)
7 sdomdomtr 8270 . . 3 (((𝐴𝐵) ≺ ω ∧ ω ≼ 𝑋) → (𝐴𝐵) ≺ 𝑋)
85, 6, 7syl2anc 574 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ 𝑋)
9 omelon 8728 . . . . . 6 ω ∈ On
10 onenon 8996 . . . . . 6 (ω ∈ On → ω ∈ dom card)
119, 10ax-mp 5 . . . . 5 ω ∈ dom card
12 simpll 772 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝑋 ∈ dom card)
13 sdomdom 8158 . . . . . . 7 (𝐵𝑋𝐵𝑋)
1413ad2antll 709 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐵𝑋)
15 numdom 9082 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐵𝑋) → 𝐵 ∈ dom card)
1612, 14, 15syl2anc 574 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → 𝐵 ∈ dom card)
17 domtri2 9036 . . . . 5 ((ω ∈ dom card ∧ 𝐵 ∈ dom card) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
1811, 16, 17sylancr 576 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
1918biimpar 464 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ¬ 𝐵 ≺ ω) → ω ≼ 𝐵)
20 uncom 3915 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
2116adantr 467 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ∈ dom card)
22 simpr 472 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → ω ≼ 𝐵)
231adantr 467 . . . . . 6 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐴𝐵)
24 infunabs 9252 . . . . . 6 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2521, 22, 23, 24syl3anc 1480 . . . . 5 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐵𝐴) ≈ 𝐵)
2620, 25syl5eqbr 4832 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐴𝐵) ≈ 𝐵)
27 simplrr 785 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → 𝐵𝑋)
28 ensdomtr 8273 . . . 4 (((𝐴𝐵) ≈ 𝐵𝐵𝑋) → (𝐴𝐵) ≺ 𝑋)
2926, 27, 28syl2anc 574 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ω ≼ 𝐵) → (𝐴𝐵) ≺ 𝑋)
3019, 29syldan 580 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) ∧ ¬ 𝐵 ≺ ω) → (𝐴𝐵) ≺ 𝑋)
318, 30pm2.61dan 836 1 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 383  wcel 2148  cun 3727   class class class wbr 4797  dom cdm 5263  Oncon0 5877  ωcom 7233  cen 8127  cdom 8128  csdm 8129  cardccrd 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-inf2 8723
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-se 5223  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-isom 6051  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-1o 7734  df-2o 7735  df-oadd 7738  df-er 7917  df-en 8131  df-dom 8132  df-sdom 8133  df-fin 8134  df-oi 8592  df-card 8986  df-cda 9213
This theorem is referenced by:  infunsdom  9259
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