Theorem harval2 10002

Description: An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
harval2	⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem harval2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		harval 9532	. . . . . . 7 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴})
2	1	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴})
3		sdomel 9122	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ≺ 𝑥 → 𝑦 ∈ 𝑥))
4		domsdomtr 9110	. . . . . . . . . . . 12 ⊢ ((𝑦 ≼ 𝐴 ∧ 𝐴 ≺ 𝑥) → 𝑦 ≺ 𝑥)
5	3, 4	impel 515	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑦 ≼ 𝐴 ∧ 𝐴 ≺ 𝑥)) → 𝑦 ∈ 𝑥)
6	5	an4s 673	. . . . . . . . . 10 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≼ 𝐴) ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → 𝑦 ∈ 𝑥)
7	6	ancoms 464	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ 𝐴 ≺ 𝑥) ∧ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴)) → 𝑦 ∈ 𝑥)
8	7	3impb 1132	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ 𝐴 ≺ 𝑥) ∧ 𝑦 ∈ On ∧ 𝑦 ≼ 𝐴) → 𝑦 ∈ 𝑥)
9	8	rabssdv 4022	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≺ 𝑥) → {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴} ⊆ 𝑥)
10	9	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴} ⊆ 𝑥)
11	2, 10	eqsstrd 3965	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → (har‘𝐴) ⊆ 𝑥)
12	11	expr 462	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
13	12	ralrimiva 3154	. . 3 ⊢ (𝐴 ∈ dom card → ∀𝑥 ∈ On (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
14		ssintrab 4931	. . 3 ⊢ ((har‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ ∀𝑥 ∈ On (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
15	13, 14	sylibr 237	. 2 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
16		breq2 5107	. . . 4 ⊢ (𝑥 = (har‘𝐴) → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ (har‘𝐴)))
17		harcl 9531	. . . . 5 ⊢ (har‘𝐴) ∈ On
18	17	a1i 11	. . . 4 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ∈ On)
19		harsdom 10000	. . . 4 ⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
20	16, 18, 19	elrabd 3647	. . 3 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
21		intss1 4923	. . 3 ⊢ ((har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ (har‘𝐴))
22	20, 21	syl 18	. 2 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ (har‘𝐴))
23	15, 22	eqssd 3948	1 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412   ⊆ wss 3899  ∩ cint 4907   class class class wbr 5103  dom cdm 5655  Oncon0 6357  ‘cfv 6533   ≼ cdom 8950   ≺ csdm 8951  harchar 9528  cardccrd 9940

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  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 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-oi 9482  df-har 9529  df-card 9944

This theorem is used by:  harsucnn  10003  alephnbtwn  10074  harval3  44378  aleph1min  44397