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Theorem cardmin2 9104
Description: The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
cardmin2 (∃𝑥 ∈ On 𝐴𝑥 ↔ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Distinct variable group:   𝑥,𝐴

Proof of Theorem cardmin2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 onintrab2 7229 . . . 4 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
21biimpi 207 . . 3 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
32adantr 468 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
4 eloni 5943 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → Ord {𝑥 ∈ On ∣ 𝐴𝑥})
5 ordelss 5949 . . . . . . . 8 ((Ord {𝑥 ∈ On ∣ 𝐴𝑥} ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
64, 5sylan 571 . . . . . . 7 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
71, 6sylanb 572 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
8 ssdomg 8235 . . . . . 6 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
93, 7, 8sylc 65 . . . . 5 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
10 onelon 5958 . . . . . . . 8 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
111, 10sylanb 572 . . . . . . 7 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
12 nfcv 2947 . . . . . . . . . . . . . 14 𝑥𝐴
13 nfcv 2947 . . . . . . . . . . . . . 14 𝑥
14 nfrab1 3310 . . . . . . . . . . . . . . 15 𝑥{𝑥 ∈ On ∣ 𝐴𝑥}
1514nfint 4675 . . . . . . . . . . . . . 14 𝑥 {𝑥 ∈ On ∣ 𝐴𝑥}
1612, 13, 15nfbr 4887 . . . . . . . . . . . . 13 𝑥 𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}
17 breq2 4844 . . . . . . . . . . . . 13 (𝑥 = {𝑥 ∈ On ∣ 𝐴𝑥} → (𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}))
1816, 17onminsb 7226 . . . . . . . . . . . 12 (∃𝑥 ∈ On 𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
19 sdomentr 8330 . . . . . . . . . . . 12 ((𝐴 {𝑥 ∈ On ∣ 𝐴𝑥} ∧ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → 𝐴𝑦)
2018, 19sylan 571 . . . . . . . . . . 11 ((∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → 𝐴𝑦)
21 breq2 4844 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2221elrab 3558 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥} ↔ (𝑦 ∈ On ∧ 𝐴𝑦))
23 ssrab2 3881 . . . . . . . . . . . . . 14 {𝑥 ∈ On ∣ 𝐴𝑥} ⊆ On
24 onnmin 7230 . . . . . . . . . . . . . 14 (({𝑥 ∈ On ∣ 𝐴𝑥} ⊆ On ∧ 𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥}) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2523, 24mpan 673 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2622, 25sylbir 226 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝐴𝑦) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2726expcom 400 . . . . . . . . . . 11 (𝐴𝑦 → (𝑦 ∈ On → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
2820, 27syl 17 . . . . . . . . . 10 ((∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → (𝑦 ∈ On → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
2928impancom 441 . . . . . . . . 9 ((∃𝑥 ∈ On 𝐴𝑥𝑦 ∈ On) → ( {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦 → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
3029con2d 131 . . . . . . . 8 ((∃𝑥 ∈ On 𝐴𝑥𝑦 ∈ On) → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦))
3130impancom 441 . . . . . . 7 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → (𝑦 ∈ On → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦))
3211, 31mpd 15 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦)
33 ensym 8238 . . . . . 6 (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦)
3432, 33nsyl 137 . . . . 5 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
35 brsdom 8212 . . . . 5 (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} ↔ (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} ∧ ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
369, 34, 35sylanbrc 574 . . . 4 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
3736ralrimiva 3153 . . 3 (∃𝑥 ∈ On 𝐴𝑥 → ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
38 iscard 9081 . . 3 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
392, 37, 38sylanbrc 574 . 2 (∃𝑥 ∈ On 𝐴𝑥 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
40 vprc 4989 . . . . . 6 ¬ V ∈ V
41 inteq 4668 . . . . . . . 8 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → {𝑥 ∈ On ∣ 𝐴𝑥} = ∅)
42 int0 4679 . . . . . . . 8 ∅ = V
4341, 42syl6eq 2855 . . . . . . 7 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → {𝑥 ∈ On ∣ 𝐴𝑥} = V)
4443eleq1d 2869 . . . . . 6 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V ↔ V ∈ V))
4540, 44mtbiri 318 . . . . 5 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V)
46 fvex 6418 . . . . . 6 (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) ∈ V
47 eleq1 2872 . . . . . 6 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) ∈ V ↔ {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V))
4846, 47mpbii 224 . . . . 5 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V)
4945, 48nsyl 137 . . . 4 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ¬ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
5049necon2ai 3006 . . 3 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ≠ ∅)
51 rabn0 4155 . . 3 ({𝑥 ∈ On ∣ 𝐴𝑥} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐴𝑥)
5250, 51sylib 209 . 2 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → ∃𝑥 ∈ On 𝐴𝑥)
5339, 52impbii 200 1 (∃𝑥 ∈ On 𝐴𝑥 ↔ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  wcel 2158  wne 2977  wral 3095  wrex 3096  {crab 3099  Vcvv 3390  wss 3766  c0 4113   cint 4665   class class class wbr 4840  Ord word 5932  Oncon0 5933  cfv 6098  cen 8186  cdom 8187  csdm 8188  cardccrd 9041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-sbc 3631  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-ord 5936  df-on 5937  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-er 7976  df-en 8190  df-dom 8191  df-sdom 8192  df-card 9045
This theorem is referenced by: (None)
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