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Theorem cardmin2 10032
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 7795 . . . 4 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
21biimpi 215 . . 3 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
32adantr 479 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
4 eloni 6375 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → Ord {𝑥 ∈ On ∣ 𝐴𝑥})
5 ordelss 6381 . . . . . . . 8 ((Ord {𝑥 ∈ On ∣ 𝐴𝑥} ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
64, 5sylan 578 . . . . . . 7 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
71, 6sylanb 579 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
8 ssdomg 9020 . . . . . 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 6390 . . . . . . . 8 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
111, 10sylanb 579 . . . . . . 7 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
12 nfcv 2892 . . . . . . . . . . . . . 14 𝑥𝐴
13 nfcv 2892 . . . . . . . . . . . . . 14 𝑥
14 nfrab1 3439 . . . . . . . . . . . . . . 15 𝑥{𝑥 ∈ On ∣ 𝐴𝑥}
1514nfint 4956 . . . . . . . . . . . . . 14 𝑥 {𝑥 ∈ On ∣ 𝐴𝑥}
1612, 13, 15nfbr 5190 . . . . . . . . . . . . 13 𝑥 𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}
17 breq2 5147 . . . . . . . . . . . . 13 (𝑥 = {𝑥 ∈ On ∣ 𝐴𝑥} → (𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}))
1816, 17onminsb 7792 . . . . . . . . . . . 12 (∃𝑥 ∈ On 𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
19 sdomentr 9138 . . . . . . . . . . . 12 ((𝐴 {𝑥 ∈ On ∣ 𝐴𝑥} ∧ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → 𝐴𝑦)
2018, 19sylan 578 . . . . . . . . . . 11 ((∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → 𝐴𝑦)
21 breq2 5147 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2221elrab 3680 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥} ↔ (𝑦 ∈ On ∧ 𝐴𝑦))
23 ssrab2 4073 . . . . . . . . . . . . . 14 {𝑥 ∈ On ∣ 𝐴𝑥} ⊆ On
24 onnmin 7796 . . . . . . . . . . . . . 14 (({𝑥 ∈ On ∣ 𝐴𝑥} ⊆ On ∧ 𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥}) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2523, 24mpan 688 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2622, 25sylbir 234 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝐴𝑦) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2726expcom 412 . . . . . . . . . . 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 450 . . . . . . . . 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 450 . . . . . . 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 9023 . . . . . 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 8995 . . . . 5 (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} ↔ (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} ∧ ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
369, 34, 35sylanbrc 581 . . . 4 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
3736ralrimiva 3136 . . 3 (∃𝑥 ∈ On 𝐴𝑥 → ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
38 iscard 10008 . . 3 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
392, 37, 38sylanbrc 581 . 2 (∃𝑥 ∈ On 𝐴𝑥 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
40 vprc 5310 . . . . . 6 ¬ V ∈ V
41 inteq 4949 . . . . . . . 8 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → {𝑥 ∈ On ∣ 𝐴𝑥} = ∅)
42 int0 4962 . . . . . . . 8 ∅ = V
4341, 42eqtrdi 2782 . . . . . . 7 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → {𝑥 ∈ On ∣ 𝐴𝑥} = V)
4443eleq1d 2811 . . . . . 6 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V ↔ V ∈ V))
4540, 44mtbiri 326 . . . . 5 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V)
46 fvex 6903 . . . . . 6 (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) ∈ V
47 eleq1 2814 . . . . . 6 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) ∈ V ↔ {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V))
4846, 47mpbii 232 . . . . 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 2960 . . 3 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ≠ ∅)
51 rabn0 4383 . . 3 ({𝑥 ∈ On ∣ 𝐴𝑥} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐴𝑥)
5250, 51sylib 217 . 2 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → ∃𝑥 ∈ On 𝐴𝑥)
5339, 52impbii 208 1 (∃𝑥 ∈ On 𝐴𝑥 ↔ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  {crab 3419  Vcvv 3462  wss 3946  c0 4322   cint 4946   class class class wbr 5143  Ord word 6364  Oncon0 6365  cfv 6543  cen 8960  cdom 8961  csdm 8962  cardccrd 9968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6368  df-on 6369  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-er 8723  df-en 8964  df-dom 8965  df-sdom 8966  df-card 9972
This theorem is referenced by: (None)
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