Theorem ackbij1lem5 9368

Description: Lemma for ackbij2 9387. (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 7352	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2		pw2eng 8341	. . . . . . 7 ⊢ (suc 𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (2o ↑𝑚 suc 𝐴))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (2o ↑𝑚 suc 𝐴))
4		df-suc 5973	. . . . . . . . 9 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5	4	oveq2i 6921	. . . . . . . 8 ⊢ (2o ↑𝑚 suc 𝐴) = (2o ↑𝑚 (𝐴 ∪ {𝐴}))
6		elex 3429	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → 𝐴 ∈ V)
7		snex 5131	. . . . . . . . . . 11 ⊢ {𝐴} ∈ V
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → {𝐴} ∈ V)
9		2onn 7992	. . . . . . . . . . . 12 ⊢ 2o ∈ ω
10	9	elexi 3430	. . . . . . . . . . 11 ⊢ 2o ∈ V
11	10	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → 2o ∈ V)
12		nnord 7339	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → Ord 𝐴)
13		orddisj 6005	. . . . . . . . . . 11 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
14	12, 13	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅)
15		mapunen 8404	. . . . . . . . . 10 ⊢ (((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 1496	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (2o ↑𝑚 (𝐴 ∪ {𝐴})) ≈ ((2o ↑𝑚 𝐴) × (2o ↑𝑚 {𝐴})))
17		ovex 6942	. . . . . . . . . . . 12 ⊢ (2o ↑𝑚 𝐴) ∈ V
18	17	enref 8261	. . . . . . . . . . 11 ⊢ (2o ↑𝑚 𝐴) ≈ (2o ↑𝑚 𝐴)
19	18	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (2o ↑𝑚 𝐴) ≈ (2o ↑𝑚 𝐴))
20		2on 7840	. . . . . . . . . . . 12 ⊢ 2o ∈ On
21	20	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → 2o ∈ On)
22		id 22	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → 𝐴 ∈ ω)
23	21, 22	mapsnend 8307	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (2o ↑𝑚 {𝐴}) ≈ 2o)
24		xpen 8398	. . . . . . . . . 10 ⊢ (((2o ↑𝑚 𝐴) ≈ (2o ↑𝑚 𝐴) ∧ (2o ↑𝑚 {𝐴}) ≈ 2o) → ((2o ↑𝑚 𝐴) × (2o ↑𝑚 {𝐴})) ≈ ((2o ↑𝑚 𝐴) × 2o))
25	19, 23, 24	syl2anc 579	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → ((2o ↑𝑚 𝐴) × (2o ↑𝑚 {𝐴})) ≈ ((2o ↑𝑚 𝐴) × 2o))
26		entr 8280	. . . . . . . . 9 ⊢ (((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))
27	16, 25, 26	syl2anc 579	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (2o ↑𝑚 (𝐴 ∪ {𝐴})) ≈ ((2o ↑𝑚 𝐴) × 2o))
28	5, 27	syl5eqbr 4910	. . . . . . 7 ⊢ (𝐴 ∈ ω → (2o ↑𝑚 suc 𝐴) ≈ ((2o ↑𝑚 𝐴) × 2o))
29		pw2eng 8341	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ≈ (2o ↑𝑚 𝐴))
30	10	enref 8261	. . . . . . . . . 10 ⊢ 2o ≈ 2o
31	30	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 2o ≈ 2o)
32		xpen 8398	. . . . . . . . 9 ⊢ ((𝒫 𝐴 ≈ (2o ↑𝑚 𝐴) ∧ 2o ≈ 2o) → (𝒫 𝐴 × 2o) ≈ ((2o ↑𝑚 𝐴) × 2o))
33	29, 31, 32	syl2anc 579	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝒫 𝐴 × 2o) ≈ ((2o ↑𝑚 𝐴) × 2o))
34	33	ensymd 8279	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((2o ↑𝑚 𝐴) × 2o) ≈ (𝒫 𝐴 × 2o))
35		entr 8280	. . . . . . 7 ⊢ (((2o ↑𝑚 suc 𝐴) ≈ ((2o ↑𝑚 𝐴) × 2o) ∧ ((2o ↑𝑚 𝐴) × 2o) ≈ (𝒫 𝐴 × 2o)) → (2o ↑𝑚 suc 𝐴) ≈ (𝒫 𝐴 × 2o))
36	28, 34, 35	syl2anc 579	. . . . . 6 ⊢ (𝐴 ∈ ω → (2o ↑𝑚 suc 𝐴) ≈ (𝒫 𝐴 × 2o))
37		entr 8280	. . . . . 6 ⊢ ((𝒫 suc 𝐴 ≈ (2o ↑𝑚 suc 𝐴) ∧ (2o ↑𝑚 suc 𝐴) ≈ (𝒫 𝐴 × 2o)) → 𝒫 suc 𝐴 ≈ (𝒫 𝐴 × 2o))
38	3, 36, 37	syl2anc 579	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (𝒫 𝐴 × 2o))
39		pwexg 5080	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ V)
40		xp2cda 9324	. . . . . 6 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 × 2o) = (𝒫 𝐴 +𝑐 𝒫 𝐴))
41	39, 40	syl 17	. . . . 5 ⊢ (𝐴 ∈ ω → (𝒫 𝐴 × 2o) = (𝒫 𝐴 +𝑐 𝒫 𝐴))
42	38, 41	breqtrd 4901	. . . 4 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴))
43		nnfi 8428	. . . . . . . 8 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
44		pwfi 8536	. . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
45	43, 44	sylib 210	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
46		ficardid 9108	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
47	45, 46	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
48		cdaen 9317	. . . . . 6 ⊢ (((card‘𝒫 𝐴) ≈ 𝒫 𝐴 ∧ (card‘𝒫 𝐴) ≈ 𝒫 𝐴) → ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)) ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴))
49	47, 47, 48	syl2anc 579	. . . . 5 ⊢ (𝐴 ∈ ω → ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)) ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴))
50	49	ensymd 8279	. . . 4 ⊢ (𝐴 ∈ ω → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)))
51		entr 8280	. . . 4 ⊢ ((𝒫 suc 𝐴 ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴) ∧ (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴))) → 𝒫 suc 𝐴 ≈ ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)))
52	42, 50, 51	syl2anc 579	. . 3 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)))
53		carden2b 9113	. . 3 ⊢ (𝒫 suc 𝐴 ≈ ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)) → (card‘𝒫 suc 𝐴) = (card‘((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴))))
54	52, 53	syl 17	. 2 ⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = (card‘((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴))))
55		ficardom 9107	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
56	45, 55	syl 17	. . 3 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
57		nnacda 9345	. . 3 ⊢ (((card‘𝒫 𝐴) ∈ ω ∧ (card‘𝒫 𝐴) ∈ ω) → (card‘((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴))) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))
58	56, 56, 57	syl2anc 579	. 2 ⊢ (𝐴 ∈ ω → (card‘((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴))) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))
59	54, 58	eqtrd 2861	1 ⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))

Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ∪ cun 3796   ∩ cin 3797  ∅c0 4146  𝒫 cpw 4380  {csn 4399   class class class wbr 4875   × cxp 5344  Ord word 5966  Oncon0 5967  suc csuc 5969  ‘cfv 6127  (class class class)co 6910  ωcom 7331  2_oc2o 7825   +_o coa 7828   ↑_𝑚 cmap 8127   ≈ cen 8225  Fincfn 8228  cardccrd 9081   +_𝑐 ccda 9311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-cda 9312

This theorem is referenced by:  ackbij1lem14  9377