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Theorem cardsdomelir 9390
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9391 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 9361 . . . 4 (card‘𝐵) ∈ On
21onelssi 6271 . . . 4 (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵))
3 ssdomg 8542 . . . 4 ((card‘𝐵) ∈ On → (𝐴 ⊆ (card‘𝐵) → 𝐴 ≼ (card‘𝐵)))
41, 2, 3mpsyl 68 . . 3 (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ (card‘𝐵))
5 elfvdm 6681 . . . 4 (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
6 cardid2 9370 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
75, 6syl 17 . . 3 (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵)
8 domentr 8555 . . 3 ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴𝐵)
94, 7, 8syl2anc 587 . 2 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
10 cardne 9382 . 2 (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)
11 brsdom 8519 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
129, 10, 11sylanbrc 586 1 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2112  wss 3884   class class class wbr 5033  dom cdm 5523  Oncon0 6163  cfv 6328  cen 8493  cdom 8494  csdm 8495  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-en 8497  df-dom 8498  df-sdom 8499  df-card 9356
This theorem is referenced by:  cardsdomel  9391  pwsdompw  9619  alephval2  9987  pwcfsdom  9998  tskcard  10196
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