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Theorem cardsdomelir 9982
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9983 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 9953 . . . 4 (cardβ€˜π΅) ∈ On
21onelssi 6478 . . . 4 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 βŠ† (cardβ€˜π΅))
3 ssdomg 9010 . . . 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 9962 . . . 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 9023 . . 3 ((𝐴 β‰Ό (cardβ€˜π΅) ∧ (cardβ€˜π΅) β‰ˆ 𝐡) β†’ 𝐴 β‰Ό 𝐡)
94, 7, 8syl2anc 583 . 2 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 β‰Ό 𝐡)
10 cardne 9974 . 2 (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)
11 brsdom 8985 . 2 (𝐴 β‰Ί 𝐡 ↔ (𝐴 β‰Ό 𝐡 ∧ Β¬ 𝐴 β‰ˆ 𝐡))
129, 10, 11sylanbrc 582 1 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 β‰Ί 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∈ wcel 2099   βŠ† wss 3944   class class class wbr 5142  dom cdm 5672  Oncon0 6363  β€˜cfv 6542   β‰ˆ cen 8950   β‰Ό cdom 8951   β‰Ί csdm 8952  cardccrd 9944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  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 5143  df-opab 5205  df-mpt 5226  df-tr 5260  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 6366  df-on 6367  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 8954  df-dom 8955  df-sdom 8956  df-card 9948
This theorem is referenced by:  cardsdomel  9983  pwsdompw  10213  alephval2  10581  pwcfsdom  10592  tskcard  10790
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