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Theorem cardonle 6794
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 6793 . 2 (𝐴 ∈ On → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
2 enrefg 6461 . . 3 (𝐴 ∈ On → 𝐴𝐴)
3 breq1 3840 . . . 4 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
43intminss 3708 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐴) → {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ 𝐴)
52, 4mpdan 412 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ 𝐴)
61, 5eqsstrd 3058 1 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438  {crab 2363  wss 2997   cint 3683   class class class wbr 3837  Oncon0 4181  cfv 5002  cen 6435  cardccrd 6786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-en 6438  df-card 6787
This theorem is referenced by:  card0  6795
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