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Theorem cardonle 7188
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 7187 . 2 (𝐴 ∈ On β†’ (cardβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
2 enrefg 6766 . . 3 (𝐴 ∈ On β†’ 𝐴 β‰ˆ 𝐴)
3 breq1 4008 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯ β‰ˆ 𝐴 ↔ 𝐴 β‰ˆ 𝐴))
43intminss 3871 . . 3 ((𝐴 ∈ On ∧ 𝐴 β‰ˆ 𝐴) β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} βŠ† 𝐴)
52, 4mpdan 421 . 2 (𝐴 ∈ On β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴} βŠ† 𝐴)
61, 5eqsstrd 3193 1 (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∈ wcel 2148  {crab 2459   βŠ† wss 3131  βˆ© cint 3846   class class class wbr 4005  Oncon0 4365  β€˜cfv 5218   β‰ˆ cen 6740  cardccrd 7180
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
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 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-en 6743  df-card 7181
This theorem is referenced by:  card0  7189
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