Theorem harval3 44155

Description: (har‘𝐴) is the least cardinal that is greater than 𝐴. (Contributed by RP, 4-Nov-2023.)

Assertion

Ref	Expression
harval3	⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem harval3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		harval2 9982	. 2 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦})
2		vex 3467	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝑥 ∈ V)
4		elrncard 44154	. . . . . . . . 9 ⊢ (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ≈ 𝑥))
5	4	simplbi 501	. . . . . . . 8 ⊢ (𝑥 ∈ ran card → 𝑥 ∈ On)
6	5	anim1i 626	. . . . . . 7 ⊢ ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥))
7		eleq1 2857	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ On ↔ 𝑥 ∈ On))
8		breq2 5117	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐴 ≺ 𝑦 ↔ 𝐴 ≺ 𝑥))
9	7, 8	anbi12d 643	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) ↔ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)))
10	6, 9	imbitrrid 249	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)))
11	10	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝑦 = 𝑥) → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)))
12		ssidd 3968	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝑥 ⊆ 𝑥)
13	3, 11, 12	intabssd 44136	. . . 4 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)} ⊆ ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)})
14		vex 3467	. . . . . . 7 ⊢ 𝑦 ∈ V
15	14	inex1 5288	. . . . . 6 ⊢ (𝑦 ∩ (card‘𝑦)) ∈ V
16	15	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ∈ V)
17		oncardid 9941	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (card‘𝑦) ≈ 𝑦)
18	17	ensymd 9001	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ≈ (card‘𝑦))
19		sdomentr 9098	. . . . . . . . . . . 12 ⊢ ((𝐴 ≺ 𝑦 ∧ 𝑦 ≈ (card‘𝑦)) → 𝐴 ≺ (card‘𝑦))
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → ((𝐴 ≺ 𝑦 ∧ 𝑦 ≈ (card‘𝑦)) → 𝐴 ≺ (card‘𝑦)))
21	18, 20	mpan2d 706	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (𝐴 ≺ 𝑦 → 𝐴 ≺ (card‘𝑦)))
22		df-card 9924	. . . . . . . . . . . 12 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
23	22	funmpt2 6576	. . . . . . . . . . 11 ⊢ Fun card
24		onenon 9934	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ∈ dom card)
25		fvelrn 7072	. . . . . . . . . . 11 ⊢ ((Fun card ∧ 𝑦 ∈ dom card) → (card‘𝑦) ∈ ran card)
26	23, 24, 25	sylancr 598	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (card‘𝑦) ∈ ran card)
27	21, 26	jctild 534	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝐴 ≺ 𝑦 → ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
28	27	adantl 486	. . . . . . . 8 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝐴 ≺ 𝑦 → ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
29		simpl 487	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (𝑦 ∩ (card‘𝑦)))
30		cardonle 9942	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (card‘𝑦) ⊆ 𝑦)
31	30	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (card‘𝑦) ⊆ 𝑦)
32		sseqin2 4184	. . . . . . . . . . 11 ⊢ ((card‘𝑦) ⊆ 𝑦 ↔ (𝑦 ∩ (card‘𝑦)) = (card‘𝑦))
33	31, 32	sylib 221	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝑦 ∩ (card‘𝑦)) = (card‘𝑦))
34	29, 33	eqtrd 2804	. . . . . . . . 9 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (card‘𝑦))
35		eleq1 2857	. . . . . . . . . 10 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ ran card ↔ (card‘𝑦) ∈ ran card))
36		breq2 5117	. . . . . . . . . 10 ⊢ (𝑥 = (card‘𝑦) → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ (card‘𝑦)))
37	35, 36	anbi12d 643	. . . . . . . . 9 ⊢ (𝑥 = (card‘𝑦) → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) ↔ ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
38	34, 37	syl 18	. . . . . . . 8 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) ↔ ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
39	28, 38	sylibrd 262	. . . . . . 7 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝐴 ≺ 𝑦 → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)))
40	39	expimpd 458	. . . . . 6 ⊢ (𝑥 = (𝑦 ∩ (card‘𝑦)) → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)))
41	40	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 = (𝑦 ∩ (card‘𝑦))) → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)))
42		inss1 4197	. . . . . 6 ⊢ (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦
43	42	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦)
44	16, 41, 43	intabssd 44136	. . . 4 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)} ⊆ ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)})
45	13, 44	eqssd 3962	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)} = ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)})
46		df-rab 3424	. . . 4 ⊢ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦} = {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)}
47	46	inteqi 4920	. . 3 ⊢ ∩ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦} = ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)}
48		df-rab 3424	. . . 4 ⊢ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)}
49	48	inteqi 4920	. . 3 ⊢ ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)}
50	45, 47, 49	3eqtr4g 2829	. 2 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦} = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})
51	1, 50	eqtrd 2804	1 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  {crab 3423  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ∩ cint 4916   class class class wbr 5113  dom cdm 5662  ran crn 5663  Oncon0 6361  Fun wfun 6531  ‘cfv 6537   ≈ cen 8939   ≺ csdm 8941  harchar 9517  cardccrd 9920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-oi 9471  df-har 9518  df-card 9924

This theorem is referenced by:  harval3on  44156