Theorem cardmin2 9990

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 7792	. . . 4 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
2	1	biimpi 219	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
3	1	birani 508	. . . . . 6 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
4		eloni 6370	. . . . . . . 8 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
5		ordelss 6376	. . . . . . . 8 ⊢ ((Ord ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
6	4, 5	sylan 591	. . . . . . 7 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
7	1, 6	sylanb 592	. . . . . 6 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
8		ssdomg 8993	. . . . . 6 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On → (𝑦 ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → 𝑦 ≼ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
9	3, 7, 8	sylc 66	. . . . 5 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ≼ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
10		onelon 6385	. . . . . . . 8 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
11	1, 10	sylanb 592	. . . . . . 7 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
12		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐴
13		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 ≺
14		nfrab1 3436	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
15	14	nfint 4922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
16	12, 13, 15	nfbr 5158	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
17		breq2 5113	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
18	16, 17	onminsb 7789	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
19		sdomentr 9095	. . . . . . . . . . . 12 ⊢ ((𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦) → 𝐴 ≺ 𝑦)
20	18, 19	sylan 591	. . . . . . . . . . 11 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦) → 𝐴 ≺ 𝑦)
21		breq2 5113	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝑦))
22	21	elrab 3650	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦))
23		ssrab2 4034	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ On
24		onnmin 7793	. . . . . . . . . . . . . 14 ⊢ (({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ On ∧ 𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
25	23, 24	mpan 702	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
26	22, 25	sylbir 238	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
27	26	expcom 418	. . . . . . . . . . 11 ⊢ (𝐴 ≺ 𝑦 → (𝑦 ∈ On → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
28	20, 27	syl 18	. . . . . . . . . 10 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦) → (𝑦 ∈ On → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
29	28	impancom 456	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ On) → (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦 → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
30	29	con2d 135	. . . . . . . 8 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ On) → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ¬ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦))
31	30	impancom 456	. . . . . . 7 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → (𝑦 ∈ On → ¬ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦))
32	11, 31	mpd 16	. . . . . 6 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ¬ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦)
33		ensym 8996	. . . . . 6 ⊢ (𝑦 ≈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≈ 𝑦)
34	32, 33	nsyl 141	. . . . 5 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ¬ 𝑦 ≈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
35		brsdom 8967	. . . . 5 ⊢ (𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (𝑦 ≼ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∧ ¬ 𝑦 ≈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
36	9, 34, 35	sylanbrc 594	. . . 4 ⊢ ((∃𝑥 ∈ On 𝐴 ≺ 𝑥 ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
37	36	ralrimiva 3157	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
38		iscard 9966	. . 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 594	. 2 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
40		vprc 5283	. . . . . 6 ⊢ ¬ V ∈ V
41		inteq 4915	. . . . . . . 8 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∩ ∅)
42		int0 4927	. . . . . . . 8 ⊢ ∩ ∅ = V
43	41, 42	eqtrdi 2814	. . . . . . 7 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = V)
44	43	eleq1d 2848	. . . . . 6 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ V ↔ V ∈ V))
45	40, 44	mtbiri 330	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ¬ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ V)
46		fvex 6894	. . . . . 6 ⊢ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) ∈ V
47		eleq1 2851	. . . . . 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 236	. . . . 5 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ V)
49	45, 48	nsyl 141	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ¬ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
50	49	necon2ai 2987	. . 3 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≠ ∅)
51		rabn0 4346	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
52	50, 51	sylib 221	. 2 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
53	39, 52	impbii 212	1 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  ∩ cint 4912   class class class wbr 5109  Ord word 6359  Oncon0 6360  ‘cfv 6536   ≈ cen 8936   ≼ cdom 8937   ≺ csdm 8938  cardccrd 9926

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-card 9930

This theorem is used by: (None)