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Theorem cardsdom2 9845
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 ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) ∈ (cardβ€˜π΅) ↔ 𝐴 β‰Ί 𝐡))

Proof of Theorem cardsdom2
StepHypRef Expression
1 carddom2 9834 . . 3 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ 𝐴 β‰Ό 𝐡))
2 carden2 9844 . . . 4 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) = (cardβ€˜π΅) ↔ 𝐴 β‰ˆ 𝐡))
32necon3abid 2977 . . 3 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) β‰  (cardβ€˜π΅) ↔ Β¬ 𝐴 β‰ˆ 𝐡))
41, 3anbi12d 631 . 2 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ (((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ∧ (cardβ€˜π΄) β‰  (cardβ€˜π΅)) ↔ (𝐴 β‰Ό 𝐡 ∧ Β¬ 𝐴 β‰ˆ 𝐡)))
5 cardon 9801 . . 3 (cardβ€˜π΄) ∈ On
6 cardon 9801 . . 3 (cardβ€˜π΅) ∈ On
7 onelpss 6342 . . 3 (((cardβ€˜π΄) ∈ On ∧ (cardβ€˜π΅) ∈ On) β†’ ((cardβ€˜π΄) ∈ (cardβ€˜π΅) ↔ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ∧ (cardβ€˜π΄) β‰  (cardβ€˜π΅))))
85, 6, 7mp2an 689 . 2 ((cardβ€˜π΄) ∈ (cardβ€˜π΅) ↔ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ∧ (cardβ€˜π΄) β‰  (cardβ€˜π΅)))
9 brsdom 8836 . 2 (𝐴 β‰Ί 𝐡 ↔ (𝐴 β‰Ό 𝐡 ∧ Β¬ 𝐴 β‰ˆ 𝐡))
104, 8, 93bitr4g 313 1 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) ∈ (cardβ€˜π΅) ↔ 𝐴 β‰Ί 𝐡))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2105   β‰  wne 2940   βŠ† wss 3898   class class class wbr 5092  dom cdm 5620  Oncon0 6302  β€˜cfv 6479   β‰ˆ cen 8801   β‰Ό cdom 8802   β‰Ί csdm 8803  cardccrd 9792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ord 6305  df-on 6306  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-card 9796
This theorem is referenced by:  domtri2  9846  nnsdomel  9847  indcardi  9898  sdom2en01  10159  cardsdom  10412  smobeth  10443  hargch  10530  cardpred  33361
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