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Theorem cardmin2 9213
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 7327 . . . 4 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
21biimpi 208 . . 3 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
32adantr 473 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
4 eloni 6033 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → Ord {𝑥 ∈ On ∣ 𝐴𝑥})
5 ordelss 6039 . . . . . . . 8 ((Ord {𝑥 ∈ On ∣ 𝐴𝑥} ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
64, 5sylan 572 . . . . . . 7 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
71, 6sylanb 573 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
8 ssdomg 8344 . . . . . 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 6048 . . . . . . . 8 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
111, 10sylanb 573 . . . . . . 7 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
12 nfcv 2926 . . . . . . . . . . . . . 14 𝑥𝐴
13 nfcv 2926 . . . . . . . . . . . . . 14 𝑥
14 nfrab1 3318 . . . . . . . . . . . . . . 15 𝑥{𝑥 ∈ On ∣ 𝐴𝑥}
1514nfint 4753 . . . . . . . . . . . . . 14 𝑥 {𝑥 ∈ On ∣ 𝐴𝑥}
1612, 13, 15nfbr 4970 . . . . . . . . . . . . 13 𝑥 𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}
17 breq2 4927 . . . . . . . . . . . . 13 (𝑥 = {𝑥 ∈ On ∣ 𝐴𝑥} → (𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}))
1816, 17onminsb 7324 . . . . . . . . . . . 12 (∃𝑥 ∈ On 𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
19 sdomentr 8439 . . . . . . . . . . . 12 ((𝐴 {𝑥 ∈ On ∣ 𝐴𝑥} ∧ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → 𝐴𝑦)
2018, 19sylan 572 . . . . . . . . . . 11 ((∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → 𝐴𝑦)
21 breq2 4927 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2221elrab 3589 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥} ↔ (𝑦 ∈ On ∧ 𝐴𝑦))
23 ssrab2 3942 . . . . . . . . . . . . . 14 {𝑥 ∈ On ∣ 𝐴𝑥} ⊆ On
24 onnmin 7328 . . . . . . . . . . . . . 14 (({𝑥 ∈ On ∣ 𝐴𝑥} ⊆ On ∧ 𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥}) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2523, 24mpan 677 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2622, 25sylbir 227 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝐴𝑦) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2726expcom 406 . . . . . . . . . . 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 444 . . . . . . . . 9 ((∃𝑥 ∈ On 𝐴𝑥𝑦 ∈ On) → ( {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦 → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
3029con2d 132 . . . . . . . 8 ((∃𝑥 ∈ On 𝐴𝑥𝑦 ∈ On) → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦))
3130impancom 444 . . . . . . 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 8347 . . . . . 6 (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦)
3432, 33nsyl 138 . . . . 5 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
35 brsdom 8321 . . . . 5 (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} ↔ (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} ∧ ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
369, 34, 35sylanbrc 575 . . . 4 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
3736ralrimiva 3126 . . 3 (∃𝑥 ∈ On 𝐴𝑥 → ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
38 iscard 9190 . . 3 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
392, 37, 38sylanbrc 575 . 2 (∃𝑥 ∈ On 𝐴𝑥 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
40 vprc 5070 . . . . . 6 ¬ V ∈ V
41 inteq 4746 . . . . . . . 8 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → {𝑥 ∈ On ∣ 𝐴𝑥} = ∅)
42 int0 4757 . . . . . . . 8 ∅ = V
4341, 42syl6eq 2824 . . . . . . 7 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → {𝑥 ∈ On ∣ 𝐴𝑥} = V)
4443eleq1d 2844 . . . . . 6 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V ↔ V ∈ V))
4540, 44mtbiri 319 . . . . 5 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V)
46 fvex 6506 . . . . . 6 (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) ∈ V
47 eleq1 2847 . . . . . 6 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) ∈ V ↔ {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V))
4846, 47mpbii 225 . . . . 5 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V)
4945, 48nsyl 138 . . . 4 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ¬ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
5049necon2ai 2990 . . 3 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ≠ ∅)
51 rabn0 4220 . . 3 ({𝑥 ∈ On ∣ 𝐴𝑥} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐴𝑥)
5250, 51sylib 210 . 2 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → ∃𝑥 ∈ On 𝐴𝑥)
5339, 52impbii 201 1 (∃𝑥 ∈ On 𝐴𝑥 ↔ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  wne 2961  wral 3082  wrex 3083  {crab 3086  Vcvv 3409  wss 3825  c0 4173   cint 4743   class class class wbr 4923  Ord word 6022  Oncon0 6023  cfv 6182  cen 8295  cdom 8296  csdm 8297  cardccrd 9150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-ord 6026  df-on 6027  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-card 9154
This theorem is referenced by: (None)
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