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Theorem infdjuabs 9463
Description: Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infdjuabs ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)

Proof of Theorem infdjuabs
StepHypRef Expression
1 simp3 1129 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
2 reldom 8353 . . . . . . 7 Rel ≼
32brrelex2i 5487 . . . . . 6 (𝐵𝐴𝐴 ∈ V)
4 djudom2 9444 . . . . . 6 ((𝐵𝐴𝐴 ∈ V) → (𝐴𝐵) ≼ (𝐴𝐴))
51, 3, 4syl2anc2 585 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴𝐴))
6 xp2dju 9437 . . . . 5 (2o × 𝐴) = (𝐴𝐴)
75, 6syl6breqr 4998 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (2o × 𝐴))
8 simp1 1127 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ∈ dom card)
9 2onn 8107 . . . . . . 7 2o ∈ ω
10 nnsdom 8952 . . . . . . 7 (2o ∈ ω → 2o ≺ ω)
11 sdomdom 8375 . . . . . . 7 (2o ≺ ω → 2o ≼ ω)
129, 10, 11mp2b 10 . . . . . 6 2o ≼ ω
13 simp2 1128 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
14 domtr 8400 . . . . . 6 ((2o ≼ ω ∧ ω ≼ 𝐴) → 2o𝐴)
1512, 13, 14sylancr 587 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 2o𝐴)
16 xpdom1g 8451 . . . . 5 ((𝐴 ∈ dom card ∧ 2o𝐴) → (2o × 𝐴) ≼ (𝐴 × 𝐴))
178, 15, 16syl2anc 584 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (2o × 𝐴) ≼ (𝐴 × 𝐴))
18 domtr 8400 . . . 4 (((𝐴𝐵) ≼ (2o × 𝐴) ∧ (2o × 𝐴) ≼ (𝐴 × 𝐴)) → (𝐴𝐵) ≼ (𝐴 × 𝐴))
197, 17, 18syl2anc 584 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴 × 𝐴))
20 infxpidm2 9278 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
21203adant3 1123 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
22 domentr 8406 . . 3 (((𝐴𝐵) ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≈ 𝐴) → (𝐴𝐵) ≼ 𝐴)
2319, 21, 22syl2anc 584 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ 𝐴)
242brrelex1i 5486 . . . 4 (𝐵𝐴𝐵 ∈ V)
25243ad2ant3 1126 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ∈ V)
26 djudoml 9445 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ V) → 𝐴 ≼ (𝐴𝐵))
278, 25, 26syl2anc 584 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
28 sbth 8474 . 2 (((𝐴𝐵) ≼ 𝐴𝐴 ≼ (𝐴𝐵)) → (𝐴𝐵) ≈ 𝐴)
2923, 27, 28syl2anc 584 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1078  wcel 2079  Vcvv 3432   class class class wbr 4956   × cxp 5433  dom cdm 5435  ωcom 7427  2oc2o 7938  cen 8344  cdom 8345  csdm 8346  cdju 9162  cardccrd 9199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-inf2 8939
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-se 5395  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-isom 6226  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-2o 7945  df-oadd 7948  df-er 8130  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-oi 8810  df-dju 9165  df-card 9203
This theorem is referenced by:  infunabs  9464  infdju  9465  infdif  9466
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