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Theorem cardsdom 9233
Description: Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
cardsdom ((𝐴𝑉𝐵𝑊) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴𝐵))

Proof of Theorem cardsdom
StepHypRef Expression
1 numth3 9152 . 2 (𝐴𝑉𝐴 ∈ dom card)
2 numth3 9152 . 2 (𝐵𝑊𝐵 ∈ dom card)
3 cardsdom2 8674 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴𝐵))
41, 2, 3syl2an 492 1 ((𝐴𝑉𝐵𝑊) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976   class class class wbr 4577  dom cdm 5028  cfv 5790  csdm 7817  cardccrd 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-ac2 9145
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-wrecs 7271  df-recs 7332  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-card 8625  df-ac 8799
This theorem is referenced by:  canth3  9239  inar1  9453
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