Theorem cardmin 10595

Description: The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
cardmin	⊢ (𝐴 ∈ 𝑉 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem cardmin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		numthcor 10525	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
2		onintrab2 7795	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
3	1, 2	sylib 217	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
4		onelon 6390	. . . . . . . . 9 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
5	4	ex 411	. . . . . . . 8 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → 𝑦 ∈ On))
6	3, 5	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → 𝑦 ∈ On))
7		breq2 5147	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝑦))
8	7	onnminsb 7797	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ¬ 𝐴 ≺ 𝑦))
9	6, 8	syli 39	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ¬ 𝐴 ≺ 𝑦))
10		vex 3466	. . . . . . 7 ⊢ 𝑦 ∈ V
11		domtri 10587	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑦 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝑦))
12	10, 11	mpan 688	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝑦))
13	9, 12	sylibrd 258	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → 𝑦 ≼ 𝐴))
14		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
15		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑥 ≺
16		nfrab1 3439	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
17	16	nfint 4956	. . . . . . . 8 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
18	14, 15, 17	nfbr 5190	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
19		breq2 5147	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
20	18, 19	onminsb 7792	. . . . . 6 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 → 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
21	1, 20	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
22	13, 21	jctird 525	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝑦 ≼ 𝐴 ∧ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})))
23		domsdomtr 9139	. . . 4 ⊢ ((𝑦 ≼ 𝐴 ∧ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
24	22, 23	syl6 35	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → 𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
25	24	ralrimiv 3135	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
26		iscard 10008	. 2 ⊢ ((card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ (∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
27	3, 25, 26	sylanbrc 581	1 ⊢ (𝐴 ∈ 𝑉 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051  ∃wrex 3060  {crab 3419  Vcvv 3462  ∩ cint 4946   class class class wbr 5143  Oncon0 6365  ‘cfv 6543   ≼ cdom 8961   ≺ csdm 8962  cardccrd 9968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735  ax-ac2 10494

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-er 8723  df-en 8964  df-dom 8965  df-sdom 8966  df-card 9972  df-ac 10149

This theorem is referenced by: (None)