Theorem harval2 9885

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 9441	. . . . . . 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 9032	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ≺ 𝑥 → 𝑦 ∈ 𝑥))
4		domsdomtr 9020	. . . . . . . . . . . 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 4020	. . . . . . 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 3964	. . . . 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 3124	. . 3 ⊢ (𝐴 ∈ dom card → ∀𝑥 ∈ On (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
14		ssintrab 4916	. . 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 5090	. . . 4 ⊢ (𝑥 = (har‘𝐴) → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ (har‘𝐴)))
17		harcl 9440	. . . . 5 ⊢ (har‘𝐴) ∈ On
18	17	a1i 11	. . . 4 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ∈ On)
19		harsdom 9883	. . . 4 ⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
20	16, 18, 19	elrabd 3644	. . 3 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
21		intss1 4908	. . 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 3947	1 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395   ⊆ wss 3897  ∩ cint 4892   class class class wbr 5086  dom cdm 5611  Oncon0 6301  ‘cfv 6476   ≼ cdom 8862   ≺ csdm 8863  harchar 9437  cardccrd 9823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7298  df-ov 7344  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-oi 9391  df-har 9438  df-card 9827

This theorem is referenced by:  harsucnn  9886  alephnbtwn  9957  harval3  43571  aleph1min  43590