Theorem cardonle 9116

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 9114	. 2 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
2		enrefg 8273	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴)
3		breq1 4889	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
4	3	intminss 4736	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ⊆ 𝐴)
5	2, 4	mpdan 677	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ⊆ 𝐴)
6	1, 5	eqsstrd 3858	1 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2107  {crab 3094   ⊆ wss 3792  ∩ cint 4710   class class class wbr 4886  Oncon0 5976  ‘cfv 6135   ≈ cen 8238  cardccrd 9094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-en 8242  df-card 9098

This theorem is referenced by:  card0  9117  iscard  9134  iscard2  9135  carduni  9140  cardom  9145  alephinit  9251  cfle  9411  cfflb  9416  pwfseqlem5  9820