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Theorem cardsdom2 9405
Description: A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cardsdom2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴𝐵))

Proof of Theorem cardsdom2
StepHypRef Expression
1 carddom2 9394 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
2 carden2 9404 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))
32necon3abid 3049 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ≠ (card‘𝐵) ↔ ¬ 𝐴𝐵))
41, 3anbi12d 630 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ (card‘𝐵)) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵)))
5 cardon 9361 . . 3 (card‘𝐴) ∈ On
6 cardon 9361 . . 3 (card‘𝐵) ∈ On
7 onelpss 6224 . . 3 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ∈ (card‘𝐵) ↔ ((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ (card‘𝐵))))
85, 6, 7mp2an 688 . 2 ((card‘𝐴) ∈ (card‘𝐵) ↔ ((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ (card‘𝐵)))
9 brsdom 8520 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
104, 8, 93bitr4g 315 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wcel 2105  wne 3013  wss 3933   class class class wbr 5057  dom cdm 5548  Oncon0 6184  cfv 6348  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-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
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-rab 3144  df-v 3494  df-sbc 3770  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-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-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-ord 6187  df-on 6188  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-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-card 9356
This theorem is referenced by:  domtri2  9406  nnsdomel  9407  indcardi  9455  sdom2en01  9712  cardsdom  9965  smobeth  9996  hargch  10083
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