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Theorem cardsdomelir 10011
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 10012 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 9982 . . . 4 (card‘𝐵) ∈ On
21onelssi 6501 . . . 4 (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵))
3 ssdomg 9039 . . . 4 ((card‘𝐵) ∈ On → (𝐴 ⊆ (card‘𝐵) → 𝐴 ≼ (card‘𝐵)))
41, 2, 3mpsyl 68 . . 3 (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ (card‘𝐵))
5 elfvdm 6944 . . . 4 (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
6 cardid2 9991 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
75, 6syl 17 . . 3 (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵)
8 domentr 9052 . . 3 ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴𝐵)
94, 7, 8syl2anc 584 . 2 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
10 cardne 10003 . 2 (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)
11 brsdom 9014 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
129, 10, 11sylanbrc 583 1 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106  wss 3963   class class class wbr 5148  dom cdm 5689  Oncon0 6386  cfv 6563  cen 8981  cdom 8982  csdm 8983  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-en 8985  df-dom 8986  df-sdom 8987  df-card 9977
This theorem is referenced by:  cardsdomel  10012  pwsdompw  10241  alephval2  10610  pwcfsdom  10621  tskcard  10819
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