Theorem cardmin2 9918

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 7746	. . . 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 6329	. . . . . . . 8 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
5		ordelss 6335	. . . . . . . 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 8942	. . . . . 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 6344	. . . . . . . 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 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐴
13		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 ≺
14		nfrab1 3410	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
15	14	nfint 4900	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
16	12, 13, 15	nfbr 5133	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
17		breq2 5090	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
18	16, 17	onminsb 7743	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
19		sdomentr 9044	. . . . . . . . . . . 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 5090	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝑦))
22	21	elrab 3635	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦))
23		ssrab2 4021	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ On
24		onnmin 7747	. . . . . . . . . . . . . 14 ⊢ (({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ On ∧ 𝑦 ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → ¬ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
25	23, 24	mpan 691	. . . . . . . . . . . . 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 8945	. . . . . 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 8916	. . . . 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 3130	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
38		iscard 9894	. . 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 5253	. . . . . 6 ⊢ ¬ V ∈ V
41		inteq 4893	. . . . . . . 8 ⊢ ({𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∅ → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} = ∩ ∅)
42		int0 4905	. . . . . . . 8 ⊢ ∩ ∅ = V
43	41, 42	eqtrdi 2788	. . . . . . 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 6849	. . . . . 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 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 2962	. . 3 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ≠ ∅)
51		rabn0 4330	. . 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 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  ∩ cint 4890   class class class wbr 5086  Ord word 6318  Oncon0 6319  ‘cfv 6494   ≈ cen 8885   ≼ cdom 8886   ≺ csdm 8887  cardccrd 9854

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ord 6322  df-on 6323  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-card 9858

This theorem is referenced by: (None)