Theorem harval3 43511

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 9893	. 2 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦})
2		vex 3440	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝑥 ∈ V)
4		elrncard 43510	. . . . . . . . 9 ⊢ (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ≈ 𝑥))
5	4	simplbi 497	. . . . . . . 8 ⊢ (𝑥 ∈ ran card → 𝑥 ∈ On)
6	5	anim1i 615	. . . . . . 7 ⊢ ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥))
7		eleq1 2816	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ On ↔ 𝑥 ∈ On))
8		breq2 5096	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐴 ≺ 𝑦 ↔ 𝐴 ≺ 𝑥))
9	7, 8	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) ↔ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)))
10	6, 9	imbitrrid 246	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)))
11	10	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝑦 = 𝑥) → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)))
12		ssidd 3959	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝑥 ⊆ 𝑥)
13	3, 11, 12	intabssd 43492	. . . 4 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)} ⊆ ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)})
14		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
15	14	inex1 5256	. . . . . 6 ⊢ (𝑦 ∩ (card‘𝑦)) ∈ V
16	15	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ∈ V)
17		oncardid 9852	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (card‘𝑦) ≈ 𝑦)
18	17	ensymd 8930	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ≈ (card‘𝑦))
19		sdomentr 9028	. . . . . . . . . . . 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 694	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (𝐴 ≺ 𝑦 → 𝐴 ≺ (card‘𝑦)))
22		df-card 9835	. . . . . . . . . . . 12 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
23	22	funmpt2 6521	. . . . . . . . . . 11 ⊢ Fun card
24		onenon 9845	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ∈ dom card)
25		fvelrn 7010	. . . . . . . . . . 11 ⊢ ((Fun card ∧ 𝑦 ∈ dom card) → (card‘𝑦) ∈ ran card)
26	23, 24, 25	sylancr 587	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (card‘𝑦) ∈ ran card)
27	21, 26	jctild 525	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝐴 ≺ 𝑦 → ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
28	27	adantl 481	. . . . . . . 8 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝐴 ≺ 𝑦 → ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
29		simpl 482	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (𝑦 ∩ (card‘𝑦)))
30		cardonle 9853	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (card‘𝑦) ⊆ 𝑦)
31	30	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (card‘𝑦) ⊆ 𝑦)
32		sseqin2 4174	. . . . . . . . . . 11 ⊢ ((card‘𝑦) ⊆ 𝑦 ↔ (𝑦 ∩ (card‘𝑦)) = (card‘𝑦))
33	31, 32	sylib 218	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝑦 ∩ (card‘𝑦)) = (card‘𝑦))
34	29, 33	eqtrd 2764	. . . . . . . . 9 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (card‘𝑦))
35		eleq1 2816	. . . . . . . . . 10 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ ran card ↔ (card‘𝑦) ∈ ran card))
36		breq2 5096	. . . . . . . . . 10 ⊢ (𝑥 = (card‘𝑦) → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ (card‘𝑦)))
37	35, 36	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = (card‘𝑦) → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) ↔ ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
38	34, 37	syl 17	. . . . . . . 8 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) ↔ ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦))))
39	28, 38	sylibrd 259	. . . . . . 7 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝐴 ≺ 𝑦 → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)))
40	39	expimpd 453	. . . . . 6 ⊢ (𝑥 = (𝑦 ∩ (card‘𝑦)) → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)))
41	40	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 = (𝑦 ∩ (card‘𝑦))) → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)))
42		inss1 4188	. . . . . 6 ⊢ (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦
43	42	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦)
44	16, 41, 43	intabssd 43492	. . . 4 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)} ⊆ ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)})
45	13, 44	eqssd 3953	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)} = ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)})
46		df-rab 3395	. . . 4 ⊢ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦} = {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)}
47	46	inteqi 4900	. . 3 ⊢ ∩ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦} = ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)}
48		df-rab 3395	. . . 4 ⊢ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)}
49	48	inteqi 4900	. . 3 ⊢ ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)}
50	45, 47, 49	3eqtr4g 2789	. 2 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦} = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})
51	1, 50	eqtrd 2764	1 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  {crab 3394  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  ∩ cint 4896   class class class wbr 5092  dom cdm 5619  ran crn 5620  Oncon0 6307  Fun wfun 6476  ‘cfv 6482   ≈ cen 8869   ≺ csdm 8871  harchar 9448  cardccrd 9831

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-oi 9402  df-har 9449  df-card 9835

This theorem is referenced by:  harval3on  43512