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Theorem cardmin2 8684
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 6871 . . . 4 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
21biimpi 204 . . 3 (∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
32adantr 479 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On)
4 eloni 5636 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → Ord {𝑥 ∈ On ∣ 𝐴𝑥})
5 ordelss 5642 . . . . . . . 8 ((Ord {𝑥 ∈ On ∣ 𝐴𝑥} ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
64, 5sylan 486 . . . . . . 7 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
71, 6sylanb 487 . . . . . 6 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
8 ssdomg 7864 . . . . . 6 ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
93, 7, 8sylc 62 . . . . 5 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
10 onelon 5651 . . . . . . . 8 (( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
111, 10sylanb 487 . . . . . . 7 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 ∈ On)
12 nfcv 2750 . . . . . . . . . . . . . 14 𝑥𝐴
13 nfcv 2750 . . . . . . . . . . . . . 14 𝑥
14 nfrab1 3098 . . . . . . . . . . . . . . 15 𝑥{𝑥 ∈ On ∣ 𝐴𝑥}
1514nfint 4415 . . . . . . . . . . . . . 14 𝑥 {𝑥 ∈ On ∣ 𝐴𝑥}
1612, 13, 15nfbr 4623 . . . . . . . . . . . . 13 𝑥 𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}
17 breq2 4581 . . . . . . . . . . . . 13 (𝑥 = {𝑥 ∈ On ∣ 𝐴𝑥} → (𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥}))
1816, 17onminsb 6868 . . . . . . . . . . . 12 (∃𝑥 ∈ On 𝐴𝑥𝐴 {𝑥 ∈ On ∣ 𝐴𝑥})
19 sdomentr 7956 . . . . . . . . . . . 12 ((𝐴 {𝑥 ∈ On ∣ 𝐴𝑥} ∧ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → 𝐴𝑦)
2018, 19sylan 486 . . . . . . . . . . 11 ((∃𝑥 ∈ On 𝐴𝑥 {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦) → 𝐴𝑦)
21 breq2 4581 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2221elrab 3330 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥} ↔ (𝑦 ∈ On ∧ 𝐴𝑦))
23 ssrab2 3649 . . . . . . . . . . . . . 14 {𝑥 ∈ On ∣ 𝐴𝑥} ⊆ On
24 onnmin 6872 . . . . . . . . . . . . . 14 (({𝑥 ∈ On ∣ 𝐴𝑥} ⊆ On ∧ 𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥}) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2523, 24mpan 701 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2622, 25sylbir 223 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝐴𝑦) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
2726expcom 449 . . . . . . . . . . 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 454 . . . . . . . . 9 ((∃𝑥 ∈ On 𝐴𝑥𝑦 ∈ On) → ( {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦 → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
3029con2d 127 . . . . . . . 8 ((∃𝑥 ∈ On 𝐴𝑥𝑦 ∈ On) → (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦))
3130impancom 454 . . . . . . 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 7868 . . . . . 6 (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ≈ 𝑦)
3432, 33nsyl 133 . . . . 5 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
35 brsdom 7841 . . . . 5 (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} ↔ (𝑦 {𝑥 ∈ On ∣ 𝐴𝑥} ∧ ¬ 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
369, 34, 35sylanbrc 694 . . . 4 ((∃𝑥 ∈ On 𝐴𝑥𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}) → 𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
3736ralrimiva 2948 . . 3 (∃𝑥 ∈ On 𝐴𝑥 → ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥})
38 iscard 8661 . . 3 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} ↔ ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ On ∧ ∀𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}𝑦 {𝑥 ∈ On ∣ 𝐴𝑥}))
392, 37, 38sylanbrc 694 . 2 (∃𝑥 ∈ On 𝐴𝑥 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
40 vprc 4719 . . . . . 6 ¬ V ∈ V
41 inteq 4407 . . . . . . . 8 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → {𝑥 ∈ On ∣ 𝐴𝑥} = ∅)
42 int0 4419 . . . . . . . 8 ∅ = V
4341, 42syl6eq 2659 . . . . . . 7 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → {𝑥 ∈ On ∣ 𝐴𝑥} = V)
4443eleq1d 2671 . . . . . 6 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ( {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V ↔ V ∈ V))
4540, 44mtbiri 315 . . . . 5 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ¬ {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V)
46 fvex 6098 . . . . . 6 (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) ∈ V
47 eleq1 2675 . . . . . 6 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) ∈ V ↔ {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V))
4846, 47mpbii 221 . . . . 5 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ∈ V)
4945, 48nsyl 133 . . . 4 ({𝑥 ∈ On ∣ 𝐴𝑥} = ∅ → ¬ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
5049necon2ai 2810 . . 3 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → {𝑥 ∈ On ∣ 𝐴𝑥} ≠ ∅)
51 rabn0 3911 . . 3 ({𝑥 ∈ On ∣ 𝐴𝑥} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐴𝑥)
5250, 51sylib 206 . 2 ((card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥} → ∃𝑥 ∈ On 𝐴𝑥)
5339, 52impbii 197 1 (∃𝑥 ∈ On 𝐴𝑥 ↔ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  {crab 2899  Vcvv 3172  wss 3539  c0 3873   cint 4404   class class class wbr 4577  Ord word 5625  Oncon0 5626  cfv 5790  cen 7815  cdom 7816  csdm 7817  cardccrd 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-card 8625
This theorem is referenced by: (None)
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