MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardsdomelir Structured version   Visualization version   GIF version

Theorem cardsdomelir 9974
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9975 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 9945 . . . 4 (card‘𝐵) ∈ On
21onelssi 6479 . . . 4 (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵))
3 ssdomg 9002 . . . 4 ((card‘𝐵) ∈ On → (𝐴 ⊆ (card‘𝐵) → 𝐴 ≼ (card‘𝐵)))
41, 2, 3mpsyl 68 . . 3 (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ (card‘𝐵))
5 elfvdm 6928 . . . 4 (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
6 cardid2 9954 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
75, 6syl 17 . . 3 (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵)
8 domentr 9015 . . 3 ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴𝐵)
94, 7, 8syl2anc 583 . 2 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
10 cardne 9966 . 2 (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)
11 brsdom 8977 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
129, 10, 11sylanbrc 582 1 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2105  wss 3948   class class class wbr 5148  dom cdm 5676  Oncon0 6364  cfv 6543  cen 8942  cdom 8943  csdm 8944  cardccrd 9936
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 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-en 8946  df-dom 8947  df-sdom 8948  df-card 9940
This theorem is referenced by:  cardsdomel  9975  pwsdompw  10205  alephval2  10573  pwcfsdom  10584  tskcard  10782
  Copyright terms: Public domain W3C validator