Theorem cardonle 7293

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 7292	. 2 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
2		enrefg 6854	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴)
3		breq1 4046	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
4	3	intminss 3909	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ⊆ 𝐴)
5	2, 4	mpdan 421	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ⊆ 𝐴)
6	1, 5	eqsstrd 3228	1 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2175  {crab 2487   ⊆ wss 3165  ∩ cint 3884   class class class wbr 4043  Oncon0 4409  ‘cfv 5270   ≈ cen 6824  cardccrd 7283

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-en 6827  df-card 7285

This theorem is referenced by:  card0  7294