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Theorem cardonle 7185
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 7184 . 2 (𝐴 ∈ On → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
2 enrefg 6763 . . 3 (𝐴 ∈ On → 𝐴𝐴)
3 breq1 4006 . . . 4 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
43intminss 3869 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐴) → {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ 𝐴)
52, 4mpdan 421 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ 𝐴)
61, 5eqsstrd 3191 1 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  {crab 2459  wss 3129   cint 3844   class class class wbr 4003  Oncon0 4363  cfv 5216  cen 6737  cardccrd 7177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-en 6740  df-card 7178
This theorem is referenced by:  card0  7186
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