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 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 7327	. . . 4 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
2	1	biimpi 208	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
3	2	adantr 473	. . . . . 6 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
4		eloni 6033	. . . . . . . 8 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
5		ordelss 6039	. . . . . . . 8 ⊢ ((Ord ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
6	4, 5	sylan 572	. . . . . . 7 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
7	1, 6	sylanb 573	. . . . . 6 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
8		ssdomg 8344	. . . . . 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 6048	. . . . . . . 8 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
11	1, 10	sylanb 573	. . . . . . 7 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
12		nfcv 2926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐴
13		nfcv 2926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 ≺
14		nfrab1 3318	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
15	14	nfint 4753	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
16	12, 13, 15	nfbr 4970	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
17		breq2 4927	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
18	16, 17	onminsb 7324	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
19		sdomentr 8439	. . . . . . . . . . . 12 ⊢ ((𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦) → 𝐴 ≺ 𝑦)
20	18, 19	sylan 572	. . . . . . . . . . 11 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦) → 𝐴 ≺ 𝑦)
21		breq2 4927	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝑦))
22	21	elrab 3589	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦))
23		ssrab2 3942	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ On
24		onnmin 7328	. . . . . . . . . . . . . 14 ⊢ (({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ On ∧ 𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
25	23, 24	mpan 677	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
26	22, 25	sylbir 227	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
27	26	expcom 406	. . . . . . . . . . 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 444	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ On) → (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦 → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
30	29	con2d 132	. . . . . . . 8 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ On) → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ¬ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦))
31	30	impancom 444	. . . . . . 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 8347	. . . . . 6 ⊢ (𝑦 ≈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦)
34	32, 33	nsyl 138	. . . . 5 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ¬ 𝑦 ≈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
35		brsdom 8321	. . . . 5 ⊢ (𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (𝑦 ≼ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ ¬ 𝑦 ≈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
36	9, 34, 35	sylanbrc 575	. . . 4 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
37	36	ralrimiva 3126	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
38		iscard 9190	. . 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 575	. 2 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
40		vprc 5070	. . . . . 6 ⊢ ¬ V ∈ V
41		inteq 4746	. . . . . . . 8 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∩ ∅)
42		int0 4757	. . . . . . . 8 ⊢ ∩ ∅ = V
43	41, 42	syl6eq 2824	. . . . . . 7 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = V)
44	43	eleq1d 2844	. . . . . 6 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ V ↔ V ∈ V))
45	40, 44	mtbiri 319	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ¬ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ V)
46		fvex 6506	. . . . . 6 ⊢ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) ∈ V
47		eleq1 2847	. . . . . 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 225	. . . . 5 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ V)
49	45, 48	nsyl 138	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ¬ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
50	49	necon2ai 2990	. . 3 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≠ ∅)
51		rabn0 4220	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
52	50, 51	sylib 210	. 2 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
53	39, 52	impbii 201	1 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

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)