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Theorem cardonle 9988
Description: The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
cardonle (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)

Proof of Theorem cardonle
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 oncardval 9986 . 2 (𝐴 ∈ On β†’ (cardβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
2 enrefg 9011 . . 3 (𝐴 ∈ On β†’ 𝐴 β‰ˆ 𝐴)
3 breq1 5155 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯ β‰ˆ 𝐴 ↔ 𝐴 β‰ˆ 𝐴))
43intminss 4981 . . 3 ((𝐴 ∈ On ∧ 𝐴 β‰ˆ 𝐴) β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} βŠ† 𝐴)
52, 4mpdan 685 . 2 (𝐴 ∈ On β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} βŠ† 𝐴)
61, 5eqsstrd 4020 1 (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098  {crab 3430   βŠ† wss 3949  βˆ© cint 4953   class class class wbr 5152  Oncon0 6374  β€˜cfv 6553   β‰ˆ cen 8967  cardccrd 9966
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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-en 8971  df-card 9970
This theorem is referenced by:  card0  9989  iscard  10006  iscard2  10007  carduni  10012  cardom  10017  alephinit  10126  cfle  10285  cfflb  10290  pwfseqlem5  10694  harval3  42999
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