Theorem cardmin2 9861

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 7715	. . . 4 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
2	1	biimpi 215	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
3	2	adantr 482	. . . . . 6 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
4		eloni 6317	. . . . . . . 8 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
5		ordelss 6323	. . . . . . . 8 ⊢ ((Ord ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
6	4, 5	sylan 581	. . . . . . 7 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
7	1, 6	sylanb 582	. . . . . 6 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
8		ssdomg 8866	. . . . . 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 6332	. . . . . . . 8 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
11	1, 10	sylanb 582	. . . . . . 7 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
12		nfcv 2905	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐴
13		nfcv 2905	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 ≺
14		nfrab1 3423	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
15	14	nfint 4909	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
16	12, 13, 15	nfbr 5144	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
17		breq2 5101	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
18	16, 17	onminsb 7712	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
19		sdomentr 8981	. . . . . . . . . . . 12 ⊢ ((𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦) → 𝐴 ≺ 𝑦)
20	18, 19	sylan 581	. . . . . . . . . . 11 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦) → 𝐴 ≺ 𝑦)
21		breq2 5101	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝑦))
22	21	elrab 3638	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦))
23		ssrab2 4029	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ On
24		onnmin 7716	. . . . . . . . . . . . . 14 ⊢ (({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ On ∧ 𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
25	23, 24	mpan 688	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
26	22, 25	sylbir 234	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
27	26	expcom 415	. . . . . . . . . . 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 453	. . . . . . . . 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 453	. . . . . . 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 8869	. . . . . 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 8841	. . . . 5 ⊢ (𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (𝑦 ≼ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ ¬ 𝑦 ≈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
36	9, 34, 35	sylanbrc 584	. . . 4 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
37	36	ralrimiva 3140	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
38		iscard 9837	. . 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 584	. 2 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
40		vprc 5264	. . . . . 6 ⊢ ¬ V ∈ V
41		inteq 4902	. . . . . . . 8 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∩ ∅)
42		int0 4915	. . . . . . . 8 ⊢ ∩ ∅ = V
43	41, 42	eqtrdi 2793	. . . . . . 7 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = V)
44	43	eleq1d 2822	. . . . . 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 6843	. . . . . 6 ⊢ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) ∈ V
47		eleq1 2825	. . . . . 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 232	. . . . 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 2971	. . 3 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≠ ∅)
51		rabn0 4337	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
52	50, 51	sylib 217	. 2 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
53	39, 52	impbii 208	1 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1541   ∈ wcel 2106   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  {crab 3404  Vcvv 3442   ⊆ wss 3902  ∅c0 4274  ∩ cint 4899   class class class wbr 5097  Ord word 6306  Oncon0 6307  ‘cfv 6484   ≈ cen 8806   ≼ cdom 8807   ≺ csdm 8808  cardccrd 9797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-int 4900  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6310  df-on 6311  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-er 8574  df-en 8810  df-dom 8811  df-sdom 8812  df-card 9801

This theorem is referenced by: (None)