Theorem ackbij1lem5 10229

Description: Lemma for ackbij2 10248. (Contributed by Stefan O'Rear, 19-Nov-2014.) (Proof shortened by AV, 18-Jul-2022.)

Assertion

Ref	Expression
ackbij1lem5	⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))

Proof of Theorem ackbij1lem5

Step	Hyp	Ref	Expression
1		peano2 7890	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2		pw2eng 9085	. . . . . . 7 ⊢ (suc 𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (2o ↑m suc 𝐴))
3	1, 2	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (2o ↑m suc 𝐴))
4		df-suc 6367	. . . . . . . . . 10 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5	4	oveq2i 7428	. . . . . . . . 9 ⊢ (2o ↑m suc 𝐴) = (2o ↑m (𝐴 ∪ {𝐴}))
6		elex 3474	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → 𝐴 ∈ V)
7		snex 5408	. . . . . . . . . . . 12 ⊢ {𝐴} ∈ V
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → {𝐴} ∈ V)
9		2onn 8634	. . . . . . . . . . . . 13 ⊢ 2o ∈ ω
10	9	elexi 3475	. . . . . . . . . . . 12 ⊢ 2o ∈ V
11	10	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → 2o ∈ V)
12		nnord 7874	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → Ord 𝐴)
13		orddisj 6400	. . . . . . . . . . . 12 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
14	12, 13	syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅)
15		mapunen 9148	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ V ∧ {𝐴} ∈ V ∧ 2o ∈ V) ∧ (𝐴 ∩ {𝐴}) = ∅) → (2o ↑m (𝐴 ∪ {𝐴})) ≈ ((2o ↑m 𝐴) × (2o ↑m {𝐴})))
16	6, 8, 11, 14, 15	syl31anc 1400	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (2o ↑m (𝐴 ∪ {𝐴})) ≈ ((2o ↑m 𝐴) × (2o ↑m {𝐴})))
17		ovex 7450	. . . . . . . . . . . 12 ⊢ (2o ↑m 𝐴) ∈ V
18	17	enref 8995	. . . . . . . . . . 11 ⊢ (2o ↑m 𝐴) ≈ (2o ↑m 𝐴)
19		2on 8473	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 2o ∈ On)
21		id 23	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 𝐴 ∈ ω)
22	20, 21	mapsnend 9047	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (2o ↑m {𝐴}) ≈ 2o)
23		xpen 9142	. . . . . . . . . . 11 ⊢ (((2o ↑m 𝐴) ≈ (2o ↑m 𝐴) ∧ (2o ↑m {𝐴}) ≈ 2o) → ((2o ↑m 𝐴) × (2o ↑m {𝐴})) ≈ ((2o ↑m 𝐴) × 2o))
24	18, 22, 23	sylancr 599	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → ((2o ↑m 𝐴) × (2o ↑m {𝐴})) ≈ ((2o ↑m 𝐴) × 2o))
25		entr 9016	. . . . . . . . . 10 ⊢ (((2o ↑m (𝐴 ∪ {𝐴})) ≈ ((2o ↑m 𝐴) × (2o ↑m {𝐴})) ∧ ((2o ↑m 𝐴) × (2o ↑m {𝐴})) ≈ ((2o ↑m 𝐴) × 2o)) → (2o ↑m (𝐴 ∪ {𝐴})) ≈ ((2o ↑m 𝐴) × 2o))
26	16, 24, 25	syl2anc 596	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (2o ↑m (𝐴 ∪ {𝐴})) ≈ ((2o ↑m 𝐴) × 2o))
27	5, 26	eqbrtrid 5144	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (2o ↑m suc 𝐴) ≈ ((2o ↑m 𝐴) × 2o))
28	17, 10	xpcomen 9070	. . . . . . . 8 ⊢ ((2o ↑m 𝐴) × 2o) ≈ (2o × (2o ↑m 𝐴))
29		entr 9016	. . . . . . . 8 ⊢ (((2o ↑m suc 𝐴) ≈ ((2o ↑m 𝐴) × 2o) ∧ ((2o ↑m 𝐴) × 2o) ≈ (2o × (2o ↑m 𝐴))) → (2o ↑m suc 𝐴) ≈ (2o × (2o ↑m 𝐴)))
30	27, 28, 29	sylancl 598	. . . . . . 7 ⊢ (𝐴 ∈ ω → (2o ↑m suc 𝐴) ≈ (2o × (2o ↑m 𝐴)))
31	10	enref 8995	. . . . . . . . 9 ⊢ 2o ≈ 2o
32		pw2eng 9085	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ≈ (2o ↑m 𝐴))
33		xpen 9142	. . . . . . . . 9 ⊢ ((2o ≈ 2o ∧ 𝒫 𝐴 ≈ (2o ↑m 𝐴)) → (2o × 𝒫 𝐴) ≈ (2o × (2o ↑m 𝐴)))
34	31, 32, 33	sylancr 599	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (2o × 𝒫 𝐴) ≈ (2o × (2o ↑m 𝐴)))
35	34	ensymd 9015	. . . . . . 7 ⊢ (𝐴 ∈ ω → (2o × (2o ↑m 𝐴)) ≈ (2o × 𝒫 𝐴))
36		entr 9016	. . . . . . 7 ⊢ (((2o ↑m suc 𝐴) ≈ (2o × (2o ↑m 𝐴)) ∧ (2o × (2o ↑m 𝐴)) ≈ (2o × 𝒫 𝐴)) → (2o ↑m suc 𝐴) ≈ (2o × 𝒫 𝐴))
37	30, 35, 36	syl2anc 596	. . . . . 6 ⊢ (𝐴 ∈ ω → (2o ↑m suc 𝐴) ≈ (2o × 𝒫 𝐴))
38		entr 9016	. . . . . 6 ⊢ ((𝒫 suc 𝐴 ≈ (2o ↑m suc 𝐴) ∧ (2o ↑m suc 𝐴) ≈ (2o × 𝒫 𝐴)) → 𝒫 suc 𝐴 ≈ (2o × 𝒫 𝐴))
39	3, 37, 38	syl2anc 596	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (2o × 𝒫 𝐴))
40		xp2dju 10183	. . . . 5 ⊢ (2o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴)
41	39, 40	breqtrdi 5150	. . . 4 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴))
42		nnfi 9166	. . . . . . . 8 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
43		pwfi 9292	. . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
44	42, 43	sylib 221	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
45		ficardid 9971	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
46	44, 45	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
47		djuen 10176	. . . . . 6 ⊢ (((card‘𝒫 𝐴) ≈ 𝒫 𝐴 ∧ (card‘𝒫 𝐴) ≈ 𝒫 𝐴) → ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)) ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴))
48	46, 46, 47	syl2anc 596	. . . . 5 ⊢ (𝐴 ∈ ω → ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)) ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴))
49	48	ensymd 9015	. . . 4 ⊢ (𝐴 ∈ ω → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)))
50		entr 9016	. . . 4 ⊢ ((𝒫 suc 𝐴 ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴) ∧ (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴))) → 𝒫 suc 𝐴 ≈ ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)))
51	41, 49, 50	syl2anc 596	. . 3 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)))
52		carden2b 9976	. . 3 ⊢ (𝒫 suc 𝐴 ≈ ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)) → (card‘𝒫 suc 𝐴) = (card‘((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴))))
53	51, 52	syl 18	. 2 ⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = (card‘((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴))))
54		ficardom 9970	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
55	44, 54	syl 18	. . 3 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
56		nnadju 10204	. . 3 ⊢ (((card‘𝒫 𝐴) ∈ ω ∧ (card‘𝒫 𝐴) ∈ ω) → (card‘((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴))) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))
57	55, 55, 56	syl2anc 596	. 2 ⊢ (𝐴 ∈ ω → (card‘((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴))) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))
58	53, 57	eqtrd 2797	1 ⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∪ cun 3900   ∩ cin 3901  ∅c0 4282  𝒫 cpw 4560  {csn 4587   class class class wbr 5107   × cxp 5657  Ord word 6360  Oncon0 6361  suc csuc 6363  ‘cfv 6537  (class class class)co 7417  ωcom 7866  2_oc2o 8453   +_o coa 8456   ↑_m cmap 8830   ≈ cen 8953  Fincfn 8956   ⊔ cdju 9907  cardccrd 9944

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-dju 9910  df-card 9948

This theorem is used by:  ackbij1lem14  10238