Theorem cardmin2 10068

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 𝐴 ≺ 𝑥 ↔ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem cardmin2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onintrab2 7833	. . . 4 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
2	1	biimpi 216	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
3	2	adantr 480	. . . . . 6 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
4		eloni 6405	. . . . . . . 8 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
5		ordelss 6411	. . . . . . . 8 ⊢ ((Ord ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
6	4, 5	sylan 579	. . . . . . 7 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
7	1, 6	sylanb 580	. . . . . 6 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
8		ssdomg 9060	. . . . . 6 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On → (𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → 𝑦 ≼ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
9	3, 7, 8	sylc 65	. . . . 5 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ≼ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
10		onelon 6420	. . . . . . . 8 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
11	1, 10	sylanb 580	. . . . . . 7 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
12		nfcv 2908	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐴
13		nfcv 2908	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 ≺
14		nfrab1 3464	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
15	14	nfint 4980	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
16	12, 13, 15	nfbr 5213	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
17		breq2 5170	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
18	16, 17	onminsb 7830	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
19		sdomentr 9177	. . . . . . . . . . . 12 ⊢ ((𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦) → 𝐴 ≺ 𝑦)
20	18, 19	sylan 579	. . . . . . . . . . 11 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦) → 𝐴 ≺ 𝑦)
21		breq2 5170	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝑦))
22	21	elrab 3708	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦))
23		ssrab2 4103	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ On
24		onnmin 7834	. . . . . . . . . . . . . 14 ⊢ (({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ On ∧ 𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
25	23, 24	mpan 689	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
26	22, 25	sylbir 235	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
27	26	expcom 413	. . . . . . . . . . 11 ⊢ (𝐴 ≺ 𝑦 → (𝑦 ∈ On → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
28	20, 27	syl 17	. . . . . . . . . 10 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦) → (𝑦 ∈ On → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
29	28	impancom 451	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ On) → (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦 → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
30	29	con2d 134	. . . . . . . 8 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ On) → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ¬ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦))
31	30	impancom 451	. . . . . . 7 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → (𝑦 ∈ On → ¬ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦))
32	11, 31	mpd 15	. . . . . 6 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ¬ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦)
33		ensym 9063	. . . . . 6 ⊢ (𝑦 ≈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦)
34	32, 33	nsyl 140	. . . . 5 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ¬ 𝑦 ≈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
35		brsdom 9035	. . . . 5 ⊢ (𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (𝑦 ≼ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ ¬ 𝑦 ≈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
36	9, 34, 35	sylanbrc 582	. . . 4 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
37	36	ralrimiva 3152	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
38		iscard 10044	. . 3 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
39	2, 37, 38	sylanbrc 582	. 2 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
40		vprc 5333	. . . . . 6 ⊢ ¬ V ∈ V
41		inteq 4973	. . . . . . . 8 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∩ ∅)
42		int0 4986	. . . . . . . 8 ⊢ ∩ ∅ = V
43	41, 42	eqtrdi 2796	. . . . . . 7 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = V)
44	43	eleq1d 2829	. . . . . 6 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ V ↔ V ∈ V))
45	40, 44	mtbiri 327	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ¬ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ V)
46		fvex 6933	. . . . . 6 ⊢ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) ∈ V
47		eleq1 2832	. . . . . 6 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) ∈ V ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ V))
48	46, 47	mpbii 233	. . . . 5 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ V)
49	45, 48	nsyl 140	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ¬ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
50	49	necon2ai 2976	. . 3 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≠ ∅)
51		rabn0 4412	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
52	50, 51	sylib 218	. 2 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
53	39, 52	impbii 209	1 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  ∩ cint 4970   class class class wbr 5166  Ord word 6394  Oncon0 6395  ‘cfv 6573   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-card 10008

This theorem is referenced by: (None)