Theorem cardmin 10488

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 10418	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
2		onintrab2 7754	. . 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 6352	. . . . . . . . 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 5104	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝑦))
8	7	onnminsb 7756	. . . . . . 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 3446	. . . . . . 7 ⊢ 𝑦 ∈ V
11		domtri 10480	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑦 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝑦))
12	10, 11	mpan 691	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝑦))
13	9, 12	sylibrd 259	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → 𝑦 ≼ 𝐴))
14		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
15		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥 ≺
16		nfrab1 3421	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
17	16	nfint 4914	. . . . . . . 8 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
18	14, 15, 17	nfbr 5147	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}
19		breq2 5104	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}))
20	18, 19	onminsb 7751	. . . . . 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 9054	. . . 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 3129	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}𝑦 ≺ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
26		iscard 9901	. 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 584	1 ⊢ (𝐴 ∈ 𝑉 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442  ∩ cint 4904   class class class wbr 5100  Oncon0 6327  ‘cfv 6502   ≼ cdom 8895   ≺ csdm 8896  cardccrd 9861

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-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-ac2 10387

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-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-card 9865  df-ac 10040

This theorem is referenced by: (None)