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

Theorem cardmin 9974
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 9904 . . 3 (𝐴𝑉 → ∃𝑥 ∈ On 𝐴𝑥)
2 onintrab2 7506 . . 3 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
31, 2sylib 219 . 2 (𝐴𝑉 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
4 onelon 6209 . . . . . . . . 9 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
54ex 413 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 ∈ On))
63, 5syl 17 . . . . . . 7 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 ∈ On))
7 breq2 5061 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
87onnminsb 7508 . . . . . . 7 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝐴𝑦))
96, 8syli 39 . . . . . 6 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝐴𝑦))
10 vex 3495 . . . . . . 7 𝑦 ∈ V
11 domtri 9966 . . . . . . 7 ((𝑦 ∈ V ∧ 𝐴𝑉) → (𝑦𝐴 ↔ ¬ 𝐴𝑦))
1210, 11mpan 686 . . . . . 6 (𝐴𝑉 → (𝑦𝐴 ↔ ¬ 𝐴𝑦))
139, 12sylibrd 260 . . . . 5 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦𝐴))
14 nfcv 2974 . . . . . . . 8 𝑥𝐴
15 nfcv 2974 . . . . . . . 8 𝑥
16 nfrab1 3382 . . . . . . . . 9 𝑥{𝑥 ∈ On ∣ 𝐴𝑥}
1716nfint 4877 . . . . . . . 8 𝑥 {𝑥 ∈ On ∣ 𝐴𝑥}
1814, 15, 17nfbr 5104 . . . . . . 7 𝑥 𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}
19 breq2 5061 . . . . . . 7 (𝑥 = {𝑥 ∈ On ∣ 𝐴𝑥} → (𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}))
2018, 19onminsb 7503 . . . . . 6 (∃𝑥 ∈ On 𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
211, 20syl 17 . . . . 5 (𝐴𝑉𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
2213, 21jctird 527 . . . 4 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → (𝑦𝐴𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})))
23 domsdomtr 8640 . . . 4 ((𝑦𝐴𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2422, 23syl6 35 . . 3 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
2524ralrimiv 3178 . 2 (𝐴𝑉 → ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
26 iscard 9392 . 2 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
273, 25, 26sylanbrc 583 1 (𝐴𝑉 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492   cint 4867   class class class wbr 5057  Oncon0 6184  cfv 6348  cdom 8495  csdm 8496  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-wrecs 7936  df-recs 7997  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-card 9356  df-ac 9530
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator