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Theorem cardsdomelir 9994
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9995 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
cardsdomelir (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 β‰Ί 𝐡)

Proof of Theorem cardsdomelir
StepHypRef Expression
1 cardon 9965 . . . 4 (cardβ€˜π΅) ∈ On
21onelssi 6479 . . . 4 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 βŠ† (cardβ€˜π΅))
3 ssdomg 9017 . . . 4 ((cardβ€˜π΅) ∈ On β†’ (𝐴 βŠ† (cardβ€˜π΅) β†’ 𝐴 β‰Ό (cardβ€˜π΅)))
41, 2, 3mpsyl 68 . . 3 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 β‰Ό (cardβ€˜π΅))
5 elfvdm 6928 . . . 4 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐡 ∈ dom card)
6 cardid2 9974 . . . 4 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
75, 6syl 17 . . 3 (𝐴 ∈ (cardβ€˜π΅) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
8 domentr 9030 . . 3 ((𝐴 β‰Ό (cardβ€˜π΅) ∧ (cardβ€˜π΅) β‰ˆ 𝐡) β†’ 𝐴 β‰Ό 𝐡)
94, 7, 8syl2anc 582 . 2 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 β‰Ό 𝐡)
10 cardne 9986 . 2 (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)
11 brsdom 8992 . 2 (𝐴 β‰Ί 𝐡 ↔ (𝐴 β‰Ό 𝐡 ∧ Β¬ 𝐴 β‰ˆ 𝐡))
129, 10, 11sylanbrc 581 1 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 β‰Ί 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∈ wcel 2098   βŠ† wss 3940   class class class wbr 5143  dom cdm 5672  Oncon0 6364  β€˜cfv 6542   β‰ˆ cen 8957   β‰Ό cdom 8958   β‰Ί csdm 8959  cardccrd 9956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-en 8961  df-dom 8962  df-sdom 8963  df-card 9960
This theorem is referenced by:  cardsdomel  9995  pwsdompw  10225  alephval2  10593  pwcfsdom  10604  tskcard  10802
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