Theorem cardonle 7532

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.

Step	Hyp	Ref	Expression
1		oncardval 7531	. 2 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
2		enrefg 7050	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴)
3		breq1 4133	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
4	3	intminss 3995	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ⊆ 𝐴)
5	2, 4	mpdan 425	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ⊆ 𝐴)
6	1, 5	eqsstrd 3284	1 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209  {crab 2532   ⊆ wss 3220  ∩ cint 3970   class class class wbr 4130  Oncon0 4508  ‘cfv 5377   ≈ cen 7020  cardccrd 7522

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-en 7023  df-card 7524

This theorem is used by:  card0  7533