Theorem cardmin 10633

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 10563	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
2		onintrab2 7833	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
3	1, 2	sylib 218	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
4		onelon 6420	. . . . . . . . 9 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
5	4	ex 412	. . . . . . . 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 5170	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝑦))
8	7	onnminsb 7835	. . . . . . 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 3492	. . . . . . 7 ⊢ 𝑦 ∈ V
11		domtri 10625	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑦 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝑦))
12	10, 11	mpan 689	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝑦))
13	9, 12	sylibrd 259	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → 𝑦 ≼ 𝐴))
14		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
15		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑥 ≺
16		nfrab1 3464	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
17	16	nfint 4980	. . . . . . . 8 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
18	14, 15, 17	nfbr 5213	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
19		breq2 5170	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
20	18, 19	onminsb 7830	. . . . . 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 526	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝑦 ≼ 𝐴 ∧ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})))
23		domsdomtr 9178	. . . 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 3151	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
26		iscard 10044	. 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 582	1 ⊢ (𝐴 ∈ 𝑉 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488  ∩ cint 4970   class class class wbr 5166  Oncon0 6395  ‘cfv 6573   ≼ cdom 9001   ≺ csdm 9002  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-ac2 10532

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-card 10008  df-ac 10185

This theorem is referenced by: (None)