Theorem cardmin 10523

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 10453	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
2		onintrab2 7782	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
3	1, 2	sylib 220	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On)
4		onelon 6373	. . . . . . . . 9 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) → 𝑦 ∈ On)
5	4	ex 416	. . . . . . . 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 5106	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝑦))
8	7	onnminsb 7784	. . . . . . 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 3460	. . . . . . 7 ⊢ 𝑦 ∈ V
11		domtri 10515	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑦 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝑦))
12	10, 11	mpan 700	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝑦))
13	9, 12	sylibrd 261	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → 𝑦 ≼ 𝐴))
14		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
15		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑥 ≺
16		nfrab1 3436	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
17	16	nfint 4917	. . . . . . . 8 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
18	14, 15, 17	nfbr 5149	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
19		breq2 5106	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
20	18, 19	onminsb 7779	. . . . . 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 534	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝑦 ≼ 𝐴 ∧ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})))
23		domsdomtr 9086	. . . 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 3155	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
26		iscard 9935	. 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 592	1 ⊢ (𝐴 ∈ 𝑉 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088  {crab 3416  Vcvv 3456  ∩ cint 4907   class class class wbr 5102  Oncon0 6348  ‘cfv 6523   ≼ cdom 8927   ≺ csdm 8928  cardccrd 9895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-ac2 10422

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-card 9899  df-ac 10074

This theorem is referenced by: (None)