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Theorem cardsdomelir 8654
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 8655 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 8625 . . . 4 (card‘𝐵) ∈ On
21onelssi 5734 . . . 4 (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵))
3 ssdomg 7859 . . . 4 ((card‘𝐵) ∈ On → (𝐴 ⊆ (card‘𝐵) → 𝐴 ≼ (card‘𝐵)))
41, 2, 3mpsyl 65 . . 3 (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ (card‘𝐵))
5 elfvdm 6110 . . . 4 (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
6 cardid2 8634 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
75, 6syl 17 . . 3 (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵)
8 domentr 7873 . . 3 ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴𝐵)
94, 7, 8syl2anc 690 . 2 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
10 cardne 8646 . 2 (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)
11 brsdom 7836 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
129, 10, 11sylanbrc 694 1 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1975  wss 3534   class class class wbr 4572  dom cdm 5023  Oncon0 5621  cfv 5785  cen 7810  cdom 7811  csdm 7812  cardccrd 8616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
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 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-ord 5624  df-on 5625  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-en 7814  df-dom 7815  df-sdom 7816  df-card 8620
This theorem is referenced by:  cardsdomel  8655  pwsdompw  8881  alephval2  9245  pwcfsdom  9256  tskcard  9454
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