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Theorem cardonle 8644
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 → (card‘𝐴) ⊆ 𝐴)

Proof of Theorem cardonle
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oncardval 8642 . 2 (𝐴 ∈ On → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
2 enrefg 7851 . . 3 (𝐴 ∈ On → 𝐴𝐴)
3 breq1 4581 . . . 4 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
43intminss 4433 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐴) → {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ 𝐴)
52, 4mpdan 699 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ 𝐴)
61, 5eqsstrd 3602 1 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  {crab 2900  wss 3540   cint 4405   class class class wbr 4578  Oncon0 5626  cfv 5790  cen 7816  cardccrd 8622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-int 4406  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-en 7820  df-card 8626
This theorem is referenced by:  card0  8645  iscard  8662  iscard2  8663  carduni  8668  cardom  8673  alephinit  8779  cfle  8937  cfflb  8942  pwfseqlem5  9342
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