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Theorem cardmin 9265
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 9199 . . 3 (𝐴𝑉 → ∃𝑥 ∈ On 𝐴𝑥)
2 onintrab2 6894 . . 3 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
31, 2sylib 207 . 2 (𝐴𝑉 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
4 onelon 5665 . . . . . . . . 9 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
54ex 449 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 ∈ On))
63, 5syl 17 . . . . . . 7 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 ∈ On))
7 breq2 4587 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
87onnminsb 6896 . . . . . . 7 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝐴𝑦))
96, 8syli 38 . . . . . 6 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝐴𝑦))
10 vex 3176 . . . . . . 7 𝑦 ∈ V
11 domtri 9257 . . . . . . 7 ((𝑦 ∈ V ∧ 𝐴𝑉) → (𝑦𝐴 ↔ ¬ 𝐴𝑦))
1210, 11mpan 702 . . . . . 6 (𝐴𝑉 → (𝑦𝐴 ↔ ¬ 𝐴𝑦))
139, 12sylibrd 248 . . . . 5 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦𝐴))
14 nfcv 2751 . . . . . . . 8 𝑥𝐴
15 nfcv 2751 . . . . . . . 8 𝑥
16 nfrab1 3099 . . . . . . . . 9 𝑥{𝑥 ∈ On ∣ 𝐴𝑥}
1716nfint 4421 . . . . . . . 8 𝑥 {𝑥 ∈ On ∣ 𝐴𝑥}
1814, 15, 17nfbr 4629 . . . . . . 7 𝑥 𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}
19 breq2 4587 . . . . . . 7 (𝑥 = {𝑥 ∈ On ∣ 𝐴𝑥} → (𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}))
2018, 19onminsb 6891 . . . . . 6 (∃𝑥 ∈ On 𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
211, 20syl 17 . . . . 5 (𝐴𝑉𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
2213, 21jctird 565 . . . 4 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → (𝑦𝐴𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})))
23 domsdomtr 7980 . . . 4 ((𝑦𝐴𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2422, 23syl6 34 . . 3 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
2524ralrimiv 2948 . 2 (𝐴𝑉 → ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
26 iscard 8684 . 2 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
273, 25, 26sylanbrc 695 1 (𝐴𝑉 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  {crab 2900  Vcvv 3173   cint 4410   class class class wbr 4583  Oncon0 5640  cfv 5804  cdom 7839  csdm 7840  cardccrd 8644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-ac2 9168
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-wrecs 7294  df-recs 7355  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-card 8648  df-ac 8822
This theorem is referenced by: (None)
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