Theorem harval3 43500

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 9926	. 2 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦})
2		vex 3448	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝑥 ∈ V)
4		elrncard 43499	. . . . . . . . 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 5106	. . . . . . . 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 3967	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝑥 ⊆ 𝑥)
13	3, 11, 12	intabssd 43481	. . . 4 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)} ⊆ ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)})
14		vex 3448	. . . . . . 7 ⊢ 𝑦 ∈ V
15	14	inex1 5267	. . . . . 6 ⊢ (𝑦 ∩ (card‘𝑦)) ∈ V
16	15	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ∈ V)
17		oncardid 9885	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (card‘𝑦) ≈ 𝑦)
18	17	ensymd 8953	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ≈ (card‘𝑦))
19		sdomentr 9052	. . . . . . . . . . . 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 9868	. . . . . . . . . . . 12 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
23	22	funmpt2 6539	. . . . . . . . . . 11 ⊢ Fun card
24		onenon 9878	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ∈ dom card)
25		fvelrn 7030	. . . . . . . . . . 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 9886	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (card‘𝑦) ⊆ 𝑦)
31	30	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (card‘𝑦) ⊆ 𝑦)
32		sseqin2 4182	. . . . . . . . . . 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 5106	. . . . . . . . . 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 4196	. . . . . 6 ⊢ (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦
43	42	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦)
44	16, 41, 43	intabssd 43481	. . . 4 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)} ⊆ ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)})
45	13, 44	eqssd 3961	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)} = ∩ {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)})
46		df-rab 3403	. . . 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 3403	. . . 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 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 3402  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  ∩ cint 4906   class class class wbr 5102  dom cdm 5631  ran crn 5632  Oncon0 6320  Fun wfun 6493  ‘cfv 6499   ≈ cen 8892   ≺ csdm 8894  harchar 9485  cardccrd 9864

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-oi 9439  df-har 9486  df-card 9868

This theorem is referenced by:  harval3on  43501