Theorem harval2 10016

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 9579	. . . . . . 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 9143	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ≺ 𝑥 → 𝑦 ∈ 𝑥))
4		domsdomtr 9131	. . . . . . . . . . . 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 4055	. . . . . . 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 3998	. . . . 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 3133	. . 3 ⊢ (𝐴 ∈ dom card → ∀𝑥 ∈ On (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
14		ssintrab 4952	. . 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 5128	. . . 4 ⊢ (𝑥 = (har‘𝐴) → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ (har‘𝐴)))
17		harcl 9578	. . . . 5 ⊢ (har‘𝐴) ∈ On
18	17	a1i 11	. . . 4 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ∈ On)
19		harsdom 10014	. . . 4 ⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
20	16, 18, 19	elrabd 3678	. . 3 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
21		intss1 4944	. . 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 3981	1 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  {crab 3420   ⊆ wss 3931  ∩ cint 4927   class class class wbr 5124  dom cdm 5659  Oncon0 6357  ‘cfv 6536   ≼ cdom 8962   ≺ csdm 8963  harchar 9575  cardccrd 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-oi 9529  df-har 9576  df-card 9958

This theorem is referenced by:  harsucnn  10017  alephnbtwn  10090  harval3  43537  aleph1min  43556