Theorem harval2 10035

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 9598	. . . . . . 7 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴})
2	1	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴})
3		sdomel 9163	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ≺ 𝑥 → 𝑦 ∈ 𝑥))
4		domsdomtr 9151	. . . . . . . . . . . 12 ⊢ ((𝑦 ≼ 𝐴 ∧ 𝐴 ≺ 𝑥) → 𝑦 ≺ 𝑥)
5	3, 4	impel 505	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑦 ≼ 𝐴 ∧ 𝐴 ≺ 𝑥)) → 𝑦 ∈ 𝑥)
6	5	an4s 660	. . . . . . . . . 10 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≼ 𝐴) ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → 𝑦 ∈ 𝑥)
7	6	ancoms 458	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ 𝐴 ≺ 𝑥) ∧ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴)) → 𝑦 ∈ 𝑥)
8	7	3impb 1114	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ 𝐴 ≺ 𝑥) ∧ 𝑦 ∈ On ∧ 𝑦 ≼ 𝐴) → 𝑦 ∈ 𝑥)
9	8	rabssdv 4085	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≺ 𝑥) → {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴} ⊆ 𝑥)
10	9	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴} ⊆ 𝑥)
11	2, 10	eqsstrd 4034	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → (har‘𝐴) ⊆ 𝑥)
12	11	expr 456	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
13	12	ralrimiva 3144	. . 3 ⊢ (𝐴 ∈ dom card → ∀𝑥 ∈ On (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
14		ssintrab 4976	. . 3 ⊢ ((har‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ ∀𝑥 ∈ On (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
15	13, 14	sylibr 234	. 2 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
16		breq2 5152	. . . 4 ⊢ (𝑥 = (har‘𝐴) → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ (har‘𝐴)))
17		harcl 9597	. . . . 5 ⊢ (har‘𝐴) ∈ On
18	17	a1i 11	. . . 4 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ∈ On)
19		harsdom 10033	. . . 4 ⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
20	16, 18, 19	elrabd 3697	. . 3 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
21		intss1 4968	. . 3 ⊢ ((har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ (har‘𝐴))
22	20, 21	syl 17	. 2 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ (har‘𝐴))
23	15, 22	eqssd 4013	1 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433   ⊆ wss 3963  ∩ cint 4951   class class class wbr 5148  dom cdm 5689  Oncon0 6386  ‘cfv 6563   ≼ cdom 8982   ≺ csdm 8983  harchar 9594  cardccrd 9973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-oi 9548  df-har 9595  df-card 9977

This theorem is referenced by:  harsucnn  10036  alephnbtwn  10109  harval3  43528  aleph1min  43547