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Theorem cardmin 9346
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 9276 . . 3 (𝐴𝑉 → ∃𝑥 ∈ On 𝐴𝑥)
2 onintrab2 6964 . . 3 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
31, 2sylib 208 . 2 (𝐴𝑉 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
4 onelon 5717 . . . . . . . . 9 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
54ex 450 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 ∈ On))
63, 5syl 17 . . . . . . 7 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 ∈ On))
7 breq2 4627 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
87onnminsb 6966 . . . . . . 7 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝐴𝑦))
96, 8syli 39 . . . . . 6 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝐴𝑦))
10 vex 3193 . . . . . . 7 𝑦 ∈ V
11 domtri 9338 . . . . . . 7 ((𝑦 ∈ V ∧ 𝐴𝑉) → (𝑦𝐴 ↔ ¬ 𝐴𝑦))
1210, 11mpan 705 . . . . . 6 (𝐴𝑉 → (𝑦𝐴 ↔ ¬ 𝐴𝑦))
139, 12sylibrd 249 . . . . 5 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦𝐴))
14 nfcv 2761 . . . . . . . 8 𝑥𝐴
15 nfcv 2761 . . . . . . . 8 𝑥
16 nfrab1 3115 . . . . . . . . 9 𝑥{𝑥 ∈ On ∣ 𝐴𝑥}
1716nfint 4458 . . . . . . . 8 𝑥 {𝑥 ∈ On ∣ 𝐴𝑥}
1814, 15, 17nfbr 4669 . . . . . . 7 𝑥 𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}
19 breq2 4627 . . . . . . 7 (𝑥 = {𝑥 ∈ On ∣ 𝐴𝑥} → (𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}))
2018, 19onminsb 6961 . . . . . 6 (∃𝑥 ∈ On 𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
211, 20syl 17 . . . . 5 (𝐴𝑉𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
2213, 21jctird 566 . . . 4 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → (𝑦𝐴𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})))
23 domsdomtr 8055 . . . 4 ((𝑦𝐴𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2422, 23syl6 35 . . 3 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
2524ralrimiv 2961 . 2 (𝐴𝑉 → ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
26 iscard 8761 . 2 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
273, 25, 26sylanbrc 697 1 (𝐴𝑉 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  {crab 2912  Vcvv 3190   cint 4447   class class class wbr 4623  Oncon0 5692  cfv 5857  cdom 7913  csdm 7914  cardccrd 8721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-ac2 9245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-wrecs 7367  df-recs 7428  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-card 8725  df-ac 8899
This theorem is referenced by: (None)
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