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Theorem cardmin2 10037
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 7817 . . . 4 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
21biimpi 216 . . 3 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
32adantr 480 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
4 eloni 6396 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → Ord {𝑥 ∈ On ∣ 𝐴𝑥})
5 ordelss 6402 . . . . . . . 8 ((Ord {𝑥 ∈ On ∣ 𝐴𝑥} ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
64, 5sylan 580 . . . . . . 7 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
71, 6sylanb 581 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
8 ssdomg 9039 . . . . . 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 6411 . . . . . . . 8 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
111, 10sylanb 581 . . . . . . 7 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
12 nfcv 2903 . . . . . . . . . . . . . 14 𝑥𝐴
13 nfcv 2903 . . . . . . . . . . . . . 14 𝑥
14 nfrab1 3454 . . . . . . . . . . . . . . 15 𝑥{𝑥 ∈ On ∣ 𝐴𝑥}
1514nfint 4961 . . . . . . . . . . . . . 14 𝑥 {𝑥 ∈ On ∣ 𝐴𝑥}
1612, 13, 15nfbr 5195 . . . . . . . . . . . . 13 𝑥 𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}
17 breq2 5152 . . . . . . . . . . . . 13 (𝑥 = {𝑥 ∈ On ∣ 𝐴𝑥} → (𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}))
1816, 17onminsb 7814 . . . . . . . . . . . 12 (∃𝑥 ∈ On 𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
19 sdomentr 9150 . . . . . . . . . . . 12 ((𝐴 {𝑥 ∈ On ∣ 𝐴𝑥} ∧ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → 𝐴𝑦)
2018, 19sylan 580 . . . . . . . . . . 11 ((∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → 𝐴𝑦)
21 breq2 5152 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2221elrab 3695 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥} ↔ (𝑦 ∈ On ∧ 𝐴𝑦))
23 ssrab2 4090 . . . . . . . . . . . . . 14 {𝑥 ∈ On ∣ 𝐴𝑥} ⊆ On
24 onnmin 7818 . . . . . . . . . . . . . 14 (({𝑥 ∈ On ∣ 𝐴𝑥} ⊆ On ∧ 𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥}) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2523, 24mpan 690 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2622, 25sylbir 235 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝐴𝑦) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2726expcom 413 . . . . . . . . . . 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 451 . . . . . . . . 9 ((∃𝑥 ∈ On 𝐴𝑥𝑦 ∈ On) → ( {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦 → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
3029con2d 134 . . . . . . . 8 ((∃𝑥 ∈ On 𝐴𝑥𝑦 ∈ On) → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦))
3130impancom 451 . . . . . . 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 9042 . . . . . 6 (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦)
3432, 33nsyl 140 . . . . 5 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
35 brsdom 9014 . . . . 5 (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} ↔ (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} ∧ ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
369, 34, 35sylanbrc 583 . . . 4 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
3736ralrimiva 3144 . . 3 (∃𝑥 ∈ On 𝐴𝑥 → ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
38 iscard 10013 . . 3 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
392, 37, 38sylanbrc 583 . 2 (∃𝑥 ∈ On 𝐴𝑥 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
40 vprc 5321 . . . . . 6 ¬ V ∈ V
41 inteq 4954 . . . . . . . 8 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → {𝑥 ∈ On ∣ 𝐴𝑥} = ∅)
42 int0 4967 . . . . . . . 8 ∅ = V
4341, 42eqtrdi 2791 . . . . . . 7 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → {𝑥 ∈ On ∣ 𝐴𝑥} = V)
4443eleq1d 2824 . . . . . 6 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V ↔ V ∈ V))
4540, 44mtbiri 327 . . . . 5 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V)
46 fvex 6920 . . . . . 6 (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) ∈ V
47 eleq1 2827 . . . . . 6 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) ∈ V ↔ {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V))
4846, 47mpbii 233 . . . . 5 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V)
4945, 48nsyl 140 . . . 4 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ¬ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
5049necon2ai 2968 . . 3 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ≠ ∅)
51 rabn0 4395 . . 3 ({𝑥 ∈ On ∣ 𝐴𝑥} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐴𝑥)
5250, 51sylib 218 . 2 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → ∃𝑥 ∈ On 𝐴𝑥)
5339, 52impbii 209 1 (∃𝑥 ∈ On 𝐴𝑥 ↔ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339   cint 4951   class class class wbr 5148  Ord word 6385  Oncon0 6386  cfv 6563  cen 8981  cdom 8982  csdm 8983  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-card 9977
This theorem is referenced by: (None)
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