MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardmin Structured version   Visualization version   GIF version

Theorem cardmin 10595
Description: The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
cardmin (𝐴𝑉 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem cardmin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 numthcor 10525 . . 3 (𝐴𝑉 → ∃𝑥 ∈ On 𝐴𝑥)
2 onintrab2 7795 . . 3 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
31, 2sylib 217 . 2 (𝐴𝑉 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
4 onelon 6390 . . . . . . . . 9 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
54ex 411 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 ∈ On))
63, 5syl 17 . . . . . . 7 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 ∈ On))
7 breq2 5147 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
87onnminsb 7797 . . . . . . 7 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝐴𝑦))
96, 8syli 39 . . . . . 6 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝐴𝑦))
10 vex 3466 . . . . . . 7 𝑦 ∈ V
11 domtri 10587 . . . . . . 7 ((𝑦 ∈ V ∧ 𝐴𝑉) → (𝑦𝐴 ↔ ¬ 𝐴𝑦))
1210, 11mpan 688 . . . . . 6 (𝐴𝑉 → (𝑦𝐴 ↔ ¬ 𝐴𝑦))
139, 12sylibrd 258 . . . . 5 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦𝐴))
14 nfcv 2892 . . . . . . . 8 𝑥𝐴
15 nfcv 2892 . . . . . . . 8 𝑥
16 nfrab1 3439 . . . . . . . . 9 𝑥{𝑥 ∈ On ∣ 𝐴𝑥}
1716nfint 4956 . . . . . . . 8 𝑥 {𝑥 ∈ On ∣ 𝐴𝑥}
1814, 15, 17nfbr 5190 . . . . . . 7 𝑥 𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}
19 breq2 5147 . . . . . . 7 (𝑥 = {𝑥 ∈ On ∣ 𝐴𝑥} → (𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}))
2018, 19onminsb 7792 . . . . . 6 (∃𝑥 ∈ On 𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
211, 20syl 17 . . . . 5 (𝐴𝑉𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
2213, 21jctird 525 . . . 4 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → (𝑦𝐴𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})))
23 domsdomtr 9139 . . . 4 ((𝑦𝐴𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2422, 23syl6 35 . . 3 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
2524ralrimiv 3135 . 2 (𝐴𝑉 → ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
26 iscard 10008 . 2 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
273, 25, 26sylanbrc 581 1 (𝐴𝑉 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  wrex 3060  {crab 3419  Vcvv 3462   cint 4946   class class class wbr 5143  Oncon0 6365  cfv 6543  cdom 8961  csdm 8962  cardccrd 9968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735  ax-ac2 10494
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-er 8723  df-en 8964  df-dom 8965  df-sdom 8966  df-card 9972  df-ac 10149
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator