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Theorem infdjuabs 9619
 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 1135 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
2 reldom 8500 . . . . . . 7 Rel ≼
32brrelex2i 5573 . . . . . 6 (𝐵𝐴𝐴 ∈ V)
4 djudom2 9596 . . . . . 6 ((𝐵𝐴𝐴 ∈ V) → (𝐴𝐵) ≼ (𝐴𝐴))
51, 3, 4syl2anc2 588 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴𝐴))
6 xp2dju 9589 . . . . 5 (2o × 𝐴) = (𝐴𝐴)
75, 6breqtrrdi 5072 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (2o × 𝐴))
8 simp1 1133 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ∈ dom card)
9 2onn 8251 . . . . . . 7 2o ∈ ω
10 nnsdom 9103 . . . . . . 7 (2o ∈ ω → 2o ≺ ω)
11 sdomdom 8522 . . . . . . 7 (2o ≺ ω → 2o ≼ ω)
129, 10, 11mp2b 10 . . . . . 6 2o ≼ ω
13 simp2 1134 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
14 domtr 8547 . . . . . 6 ((2o ≼ ω ∧ ω ≼ 𝐴) → 2o𝐴)
1512, 13, 14sylancr 590 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 2o𝐴)
16 xpdom1g 8599 . . . . 5 ((𝐴 ∈ dom card ∧ 2o𝐴) → (2o × 𝐴) ≼ (𝐴 × 𝐴))
178, 15, 16syl2anc 587 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (2o × 𝐴) ≼ (𝐴 × 𝐴))
18 domtr 8547 . . . 4 (((𝐴𝐵) ≼ (2o × 𝐴) ∧ (2o × 𝐴) ≼ (𝐴 × 𝐴)) → (𝐴𝐵) ≼ (𝐴 × 𝐴))
197, 17, 18syl2anc 587 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴 × 𝐴))
20 infxpidm2 9430 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
21203adant3 1129 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
22 domentr 8553 . . 3 (((𝐴𝐵) ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≈ 𝐴) → (𝐴𝐵) ≼ 𝐴)
2319, 21, 22syl2anc 587 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≼ 𝐴)
242brrelex1i 5572 . . . 4 (𝐵𝐴𝐵 ∈ V)
25243ad2ant3 1132 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ∈ V)
26 djudoml 9597 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ V) → 𝐴 ≼ (𝐴𝐵))
278, 25, 26syl2anc 587 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≼ (𝐴𝐵))
28 sbth 8623 . 2 (((𝐴𝐵) ≼ 𝐴𝐴 ≼ (𝐴𝐵)) → (𝐴𝐵) ≈ 𝐴)
2923, 27, 28syl2anc 587 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   ∈ wcel 2111  Vcvv 3441   class class class wbr 5030   × cxp 5517  dom cdm 5519  ωcom 7562  2oc2o 8081   ≈ cen 8491   ≼ cdom 8492   ≺ csdm 8493   ⊔ cdju 9313  cardccrd 9350 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-inf2 9090 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-2o 8088  df-oadd 8091  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-oi 8960  df-dju 9316  df-card 9354 This theorem is referenced by:  infunabs  9620  infdju  9621  infdif  9622
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