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Theorem cardsdomelir 9194
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9195 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
cardsdomelir (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)

Proof of Theorem cardsdomelir
StepHypRef Expression
1 cardon 9165 . . . 4 (card‘𝐵) ∈ On
21onelssi 6134 . . . 4 (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵))
3 ssdomg 8350 . . . 4 ((card‘𝐵) ∈ On → (𝐴 ⊆ (card‘𝐵) → 𝐴 ≼ (card‘𝐵)))
41, 2, 3mpsyl 68 . . 3 (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ (card‘𝐵))
5 elfvdm 6528 . . . 4 (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
6 cardid2 9174 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
75, 6syl 17 . . 3 (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵)
8 domentr 8363 . . 3 ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴𝐵)
94, 7, 8syl2anc 576 . 2 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
10 cardne 9186 . 2 (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)
11 brsdom 8327 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
129, 10, 11sylanbrc 575 1 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2051  wss 3822   class class class wbr 4925  dom cdm 5403  Oncon0 6026  cfv 6185  cen 8301  cdom 8302  csdm 8303  cardccrd 9156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-ord 6029  df-on 6030  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-en 8305  df-dom 8306  df-sdom 8307  df-card 9160
This theorem is referenced by:  cardsdomel  9195  pwsdompw  9422  alephval2  9790  pwcfsdom  9801  tskcard  9999
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