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Theorem cardmin 9588
 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 9518 . . 3 (𝐴𝑉 → ∃𝑥 ∈ On 𝐴𝑥)
2 onintrab2 7149 . . 3 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
31, 2sylib 208 . 2 (𝐴𝑉 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
4 onelon 5891 . . . . . . . . 9 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
54ex 397 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 ∈ On))
63, 5syl 17 . . . . . . 7 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 ∈ On))
7 breq2 4790 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
87onnminsb 7151 . . . . . . 7 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝐴𝑦))
96, 8syli 39 . . . . . 6 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝐴𝑦))
10 vex 3354 . . . . . . 7 𝑦 ∈ V
11 domtri 9580 . . . . . . 7 ((𝑦 ∈ V ∧ 𝐴𝑉) → (𝑦𝐴 ↔ ¬ 𝐴𝑦))
1210, 11mpan 670 . . . . . 6 (𝐴𝑉 → (𝑦𝐴 ↔ ¬ 𝐴𝑦))
139, 12sylibrd 249 . . . . 5 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦𝐴))
14 nfcv 2913 . . . . . . . 8 𝑥𝐴
15 nfcv 2913 . . . . . . . 8 𝑥
16 nfrab1 3271 . . . . . . . . 9 𝑥{𝑥 ∈ On ∣ 𝐴𝑥}
1716nfint 4621 . . . . . . . 8 𝑥 {𝑥 ∈ On ∣ 𝐴𝑥}
1814, 15, 17nfbr 4833 . . . . . . 7 𝑥 𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}
19 breq2 4790 . . . . . . 7 (𝑥 = {𝑥 ∈ On ∣ 𝐴𝑥} → (𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}))
2018, 19onminsb 7146 . . . . . 6 (∃𝑥 ∈ On 𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
211, 20syl 17 . . . . 5 (𝐴𝑉𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
2213, 21jctird 516 . . . 4 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → (𝑦𝐴𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})))
23 domsdomtr 8251 . . . 4 ((𝑦𝐴𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2422, 23syl6 35 . . 3 (𝐴𝑉 → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
2524ralrimiv 3114 . 2 (𝐴𝑉 → ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
26 iscard 9001 . 2 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
273, 25, 26sylanbrc 572 1 (𝐴𝑉 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3351  ∩ cint 4611   class class class wbr 4786  Oncon0 5866  ‘cfv 6031   ≼ cdom 8107   ≺ csdm 8108  cardccrd 8961 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-ac2 9487 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-wrecs 7559  df-recs 7621  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-card 8965  df-ac 9139 This theorem is referenced by: (None)
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