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Theorem infdif2 9620
Description: Cardinality ordering for an infinite class difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infdif2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ≼ 𝐵𝐴𝐵))

Proof of Theorem infdif2
StepHypRef Expression
1 domnsym 8631 . . . . . . 7 ((𝐴𝐵) ≼ 𝐵 → ¬ 𝐵 ≺ (𝐴𝐵))
2 simp3 1130 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
3 infdif 9619 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
43ensymd 8548 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≈ (𝐴𝐵))
5 sdomentr 8639 . . . . . . . 8 ((𝐵𝐴𝐴 ≈ (𝐴𝐵)) → 𝐵 ≺ (𝐴𝐵))
62, 4, 5syl2anc 584 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ≺ (𝐴𝐵))
71, 6nsyl3 140 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ¬ (𝐴𝐵) ≼ 𝐵)
873expia 1113 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐵𝐴 → ¬ (𝐴𝐵) ≼ 𝐵))
983adant2 1123 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐵𝐴 → ¬ (𝐴𝐵) ≼ 𝐵))
109con2d 136 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ≼ 𝐵 → ¬ 𝐵𝐴))
11 domtri2 9406 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
12113adant3 1124 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1310, 12sylibrd 260 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ≼ 𝐵𝐴𝐵))
14 simp1 1128 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
15 difss 4105 . . . 4 (𝐴𝐵) ⊆ 𝐴
16 ssdomg 8543 . . . 4 (𝐴 ∈ dom card → ((𝐴𝐵) ⊆ 𝐴 → (𝐴𝐵) ≼ 𝐴))
1714, 15, 16mpisyl 21 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ≼ 𝐴)
18 domtr 8550 . . . 4 (((𝐴𝐵) ≼ 𝐴𝐴𝐵) → (𝐴𝐵) ≼ 𝐵)
1918ex 413 . . 3 ((𝐴𝐵) ≼ 𝐴 → (𝐴𝐵 → (𝐴𝐵) ≼ 𝐵))
2017, 19syl 17 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵 → (𝐴𝐵) ≼ 𝐵))
2113, 20impbid 213 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ≼ 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  w3a 1079  wcel 2105  cdif 3930  wss 3933   class class class wbr 5057  dom cdm 5548  ωcom 7569  cen 8494  cdom 8495  csdm 8496  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-dju 9318  df-card 9356
This theorem is referenced by:  axgroth3  10241
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