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Theorem cardmin2 9940
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 (βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ↔ (cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem cardmin2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 onintrab2 7733 . . . 4 (βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ↔ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ On)
21biimpi 215 . . 3 (βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ On)
32adantr 482 . . . . . 6 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ On)
4 eloni 6328 . . . . . . . 8 (∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ On β†’ Ord ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
5 ordelss 6334 . . . . . . . 8 ((Ord ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ 𝑦 βŠ† ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
64, 5sylan 581 . . . . . . 7 ((∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ 𝑦 βŠ† ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
71, 6sylanb 582 . . . . . 6 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ 𝑦 βŠ† ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
8 ssdomg 8943 . . . . . 6 (∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ On β†’ (𝑦 βŠ† ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ 𝑦 β‰Ό ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}))
93, 7, 8sylc 65 . . . . 5 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ 𝑦 β‰Ό ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
10 onelon 6343 . . . . . . . 8 ((∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ 𝑦 ∈ On)
111, 10sylanb 582 . . . . . . 7 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ 𝑦 ∈ On)
12 nfcv 2904 . . . . . . . . . . . . . 14 β„²π‘₯𝐴
13 nfcv 2904 . . . . . . . . . . . . . 14 β„²π‘₯ β‰Ί
14 nfrab1 3425 . . . . . . . . . . . . . . 15 β„²π‘₯{π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}
1514nfint 4918 . . . . . . . . . . . . . 14 β„²π‘₯∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}
1612, 13, 15nfbr 5153 . . . . . . . . . . . . 13 β„²π‘₯ 𝐴 β‰Ί ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}
17 breq2 5110 . . . . . . . . . . . . 13 (π‘₯ = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ (𝐴 β‰Ί π‘₯ ↔ 𝐴 β‰Ί ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}))
1816, 17onminsb 7730 . . . . . . . . . . . 12 (βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ β†’ 𝐴 β‰Ί ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
19 sdomentr 9058 . . . . . . . . . . . 12 ((𝐴 β‰Ί ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β‰ˆ 𝑦) β†’ 𝐴 β‰Ί 𝑦)
2018, 19sylan 581 . . . . . . . . . . 11 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β‰ˆ 𝑦) β†’ 𝐴 β‰Ί 𝑦)
21 breq2 5110 . . . . . . . . . . . . . 14 (π‘₯ = 𝑦 β†’ (𝐴 β‰Ί π‘₯ ↔ 𝐴 β‰Ί 𝑦))
2221elrab 3646 . . . . . . . . . . . . 13 (𝑦 ∈ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ↔ (𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦))
23 ssrab2 4038 . . . . . . . . . . . . . 14 {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} βŠ† On
24 onnmin 7734 . . . . . . . . . . . . . 14 (({π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} βŠ† On ∧ 𝑦 ∈ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ Β¬ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
2523, 24mpan 689 . . . . . . . . . . . . 13 (𝑦 ∈ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ Β¬ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
2622, 25sylbir 234 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝐴 β‰Ί 𝑦) β†’ Β¬ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
2726expcom 415 . . . . . . . . . . 11 (𝐴 β‰Ί 𝑦 β†’ (𝑦 ∈ On β†’ Β¬ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}))
2820, 27syl 17 . . . . . . . . . 10 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β‰ˆ 𝑦) β†’ (𝑦 ∈ On β†’ Β¬ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}))
2928impancom 453 . . . . . . . . 9 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ 𝑦 ∈ On) β†’ (∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β‰ˆ 𝑦 β†’ Β¬ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}))
3029con2d 134 . . . . . . . 8 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ 𝑦 ∈ On) β†’ (𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ Β¬ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β‰ˆ 𝑦))
3130impancom 453 . . . . . . 7 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ (𝑦 ∈ On β†’ Β¬ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β‰ˆ 𝑦))
3211, 31mpd 15 . . . . . 6 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ Β¬ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β‰ˆ 𝑦)
33 ensym 8946 . . . . . 6 (𝑦 β‰ˆ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β‰ˆ 𝑦)
3432, 33nsyl 140 . . . . 5 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ Β¬ 𝑦 β‰ˆ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
35 brsdom 8918 . . . . 5 (𝑦 β‰Ί ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ↔ (𝑦 β‰Ό ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∧ Β¬ 𝑦 β‰ˆ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}))
369, 34, 35sylanbrc 584 . . . 4 ((βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) β†’ 𝑦 β‰Ί ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
3736ralrimiva 3140 . . 3 (βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ β†’ βˆ€π‘¦ ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}𝑦 β‰Ί ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
38 iscard 9916 . . 3 ((cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ↔ (∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ On ∧ βˆ€π‘¦ ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}𝑦 β‰Ί ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}))
392, 37, 38sylanbrc 584 . 2 (βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ β†’ (cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
40 vprc 5273 . . . . . 6 Β¬ V ∈ V
41 inteq 4911 . . . . . . . 8 ({π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} = βˆ… β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} = ∩ βˆ…)
42 int0 4924 . . . . . . . 8 ∩ βˆ… = V
4341, 42eqtrdi 2789 . . . . . . 7 ({π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} = βˆ… β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} = V)
4443eleq1d 2819 . . . . . 6 ({π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} = βˆ… β†’ (∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ V ↔ V ∈ V))
4540, 44mtbiri 327 . . . . 5 ({π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} = βˆ… β†’ Β¬ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ V)
46 fvex 6856 . . . . . 6 (cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) ∈ V
47 eleq1 2822 . . . . . 6 ((cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ ((cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) ∈ V ↔ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ V))
4846, 47mpbii 232 . . . . 5 ((cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ∈ V)
4945, 48nsyl 140 . . . 4 ({π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} = βˆ… β†’ Β¬ (cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
5049necon2ai 2970 . . 3 ((cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β‰  βˆ…)
51 rabn0 4346 . . 3 ({π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β‰  βˆ… ↔ βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯)
5250, 51sylib 217 . 2 ((cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯)
5339, 52impbii 208 1 (βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯ ↔ (cardβ€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯}) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406  Vcvv 3444   βŠ† wss 3911  βˆ…c0 4283  βˆ© cint 4908   class class class wbr 5106  Ord word 6317  Oncon0 6318  β€˜cfv 6497   β‰ˆ cen 8883   β‰Ό cdom 8884   β‰Ί csdm 8885  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-card 9880
This theorem is referenced by: (None)
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