Theorem harval3 44676

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 10085	. 2 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦})
2		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝑥 ∈ V)
4		elrncard 44675	. . . . . . . . 9 ⊢ (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ≈ 𝑥))
5	4	simplbi 502	. . . . . . . 8 ⊢ (𝑥 ∈ ran card → 𝑥 ∈ On)
6	5	anim1i 627	. . . . . . 7 ⊢ ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥))
7		eleq1 2848	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ On ↔ 𝑥 ∈ On))
8		breq2 5106	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐴 ≺ 𝑦 ↔ 𝐴 ≺ 𝑥))
9	7, 8	anbi12d 644	. . . . . . 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 487	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝑦 = 𝑥) → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)))
12		ssidd 3932	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝑥 ⊆ 𝑥)
13	3, 11, 12	intabssd 44657	. . . 4 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)} ⊆ ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)})
14		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
15	14	inex1 5277	. . . . . 6 ⊢ (𝑦 ∩ (card‘𝑦)) ∈ V
16	15	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ∈ V)
17		oncardid 10044	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (card‘𝑦) ≈ 𝑦)
18	17	ensymd 9040	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ≈ (card‘𝑦))
19		sdomentr 9138	. . . . . . . . . . . 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 707	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (𝐴 ≺ 𝑦 → 𝐴 ≺ (card‘𝑦)))
22		df-card 10027	. . . . . . . . . . . 12 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
23	22	funmpt2 6584	. . . . . . . . . . 11 ⊢ Fun card
24		onenon 10037	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ∈ dom card)
25		fvelrn 7081	. . . . . . . . . . 11 ⊢ ((Fun card ∧ 𝑦 ∈ dom card) → (card‘𝑦) ∈ ran card)
26	23, 24, 25	sylancr 599	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (card‘𝑦) ∈ ran card)
27	21, 26	jctild 535	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝐴 ≺ 𝑦 → ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
28	27	adantl 487	. . . . . . . 8 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝐴 ≺ 𝑦 → ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
29		simpl 488	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (𝑦 ∩ (card‘𝑦)))
30		cardonle 10045	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (card‘𝑦) ⊆ 𝑦)
31	30	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (card‘𝑦) ⊆ 𝑦)
32		sseqin2 4119	. . . . . . . . . . 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 2795	. . . . . . . . 9 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (card‘𝑦))
35		eleq1 2848	. . . . . . . . . 10 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ ran card ↔ (card‘𝑦) ∈ ran card))
36		breq2 5106	. . . . . . . . . 10 ⊢ (𝑥 = (card‘𝑦) → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ (card‘𝑦)))
37	35, 36	anbi12d 644	. . . . . . . . 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 459	. . . . . 6 ⊢ (𝑥 = (𝑦 ∩ (card‘𝑦)) → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)))
41	40	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 = (𝑦 ∩ (card‘𝑦))) → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)))
42		inss1 4132	. . . . . 6 ⊢ (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦
43	42	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦)
44	16, 41, 43	intabssd 44657	. . . 4 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)} ⊆ ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)})
45	13, 44	eqssd 3926	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)} = ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)})
46		df-rab 3413	. . . 4 ⊢ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦} = {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)}
47	46	inteqi 4910	. . 3 ⊢ ∩ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦} = ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)}
48		df-rab 3413	. . . 4 ⊢ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)}
49	48	inteqi 4910	. . 3 ⊢ ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)}
50	45, 47, 49	3eqtr4g 2820	. 2 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦} = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})
51	1, 50	eqtrd 2795	1 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088  ∩ cint 4906   class class class wbr 5102  dom cdm 5651  ran crn 5652  Oncon0 6365  Fun wfun 6537  ‘cfv 6543   ≈ cen 8978   ≺ csdm 8980  harchar 9557  cardccrd 10023

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-er 8725  df-en 8982  df-dom 8983  df-sdom 8984  df-oi 9511  df-har 9558  df-card 10027

This theorem is used by:  harval3on  44677