Theorem ackbij1lem5 9717

Description: Lemma for ackbij2 9736. (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 7615	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2		pw2eng 8665	. . . . . . 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 6172	. . . . . . . . . 10 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5	4	oveq2i 7175	. . . . . . . . 9 ⊢ (2o ↑m suc 𝐴) = (2o ↑m (𝐴 ∪ {𝐴}))
6		elex 3415	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → 𝐴 ∈ V)
7		snex 5295	. . . . . . . . . . . 12 ⊢ {𝐴} ∈ V
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → {𝐴} ∈ V)
9		2onn 8290	. . . . . . . . . . . . 13 ⊢ 2o ∈ ω
10	9	elexi 3416	. . . . . . . . . . . 12 ⊢ 2o ∈ V
11	10	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → 2o ∈ V)
12		nnord 7601	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → Ord 𝐴)
13		orddisj 6204	. . . . . . . . . . . 12 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅)
15		mapunen 8729	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (2o ↑m (𝐴 ∪ {𝐴})) ≈ ((2o ↑m 𝐴) × (2o ↑m {𝐴})))
17		ovex 7197	. . . . . . . . . . . 12 ⊢ (2o ↑m 𝐴) ∈ V
18	17	enref 8581	. . . . . . . . . . 11 ⊢ (2o ↑m 𝐴) ≈ (2o ↑m 𝐴)
19		2on 8132	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 2o ∈ On)
21		id 22	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 𝐴 ∈ ω)
22	20, 21	mapsnend 8628	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (2o ↑m {𝐴}) ≈ 2o)
23		xpen 8723	. . . . . . . . . . 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 590	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → ((2o ↑m 𝐴) × (2o ↑m {𝐴})) ≈ ((2o ↑m 𝐴) × 2o))
25		entr 8600	. . . . . . . . . 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 587	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (2o ↑m (𝐴 ∪ {𝐴})) ≈ ((2o ↑m 𝐴) × 2o))
27	5, 26	eqbrtrid 5062	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (2o ↑m suc 𝐴) ≈ ((2o ↑m 𝐴) × 2o))
28	17, 10	xpcomen 8650	. . . . . . . 8 ⊢ ((2o ↑m 𝐴) × 2o) ≈ (2o × (2o ↑m 𝐴))
29		entr 8600	. . . . . . . 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 589	. . . . . . 7 ⊢ (𝐴 ∈ ω → (2o ↑m suc 𝐴) ≈ (2o × (2o ↑m 𝐴)))
31	10	enref 8581	. . . . . . . . 9 ⊢ 2o ≈ 2o
32		pw2eng 8665	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ≈ (2o ↑m 𝐴))
33		xpen 8723	. . . . . . . . 9 ⊢ ((2o ≈ 2o ∧ 𝒫 𝐴 ≈ (2o ↑m 𝐴)) → (2o × 𝒫 𝐴) ≈ (2o × (2o ↑m 𝐴)))
34	31, 32, 33	sylancr 590	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (2o × 𝒫 𝐴) ≈ (2o × (2o ↑m 𝐴)))
35	34	ensymd 8599	. . . . . . 7 ⊢ (𝐴 ∈ ω → (2o × (2o ↑m 𝐴)) ≈ (2o × 𝒫 𝐴))
36		entr 8600	. . . . . . 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 587	. . . . . 6 ⊢ (𝐴 ∈ ω → (2o ↑m suc 𝐴) ≈ (2o × 𝒫 𝐴))
38		entr 8600	. . . . . 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 587	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (2o × 𝒫 𝐴))
40		xp2dju 9669	. . . . 5 ⊢ (2o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴)
41	39, 40	breqtrdi 5068	. . . 4 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴))
42		nnfi 8759	. . . . . . . 8 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
43		pwfi 8769	. . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
44	42, 43	sylib 221	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
45		ficardid 9457	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
46	44, 45	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
47		djuen 9662	. . . . . 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 587	. . . . 5 ⊢ (𝐴 ∈ ω → ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)) ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴))
49	48	ensymd 8599	. . . 4 ⊢ (𝐴 ∈ ω → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)))
50		entr 8600	. . . 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 587	. . 3 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)))
52		carden2b 9462	. . 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 17	. 2 ⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = (card‘((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴))))
54		ficardom 9456	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
55	44, 54	syl 17	. . 3 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
56		nnadju 9690	. . 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 587	. 2 ⊢ (𝐴 ∈ ω → (card‘((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴))) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))
58	53, 57	eqtrd 2773	1 ⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2113  Vcvv 3397   ∪ cun 3839   ∩ cin 3840  ∅c0 4209  𝒫 cpw 4485  {csn 4513   class class class wbr 5027   × cxp 5517  Ord word 6165  Oncon0 6166  suc csuc 6168  ‘cfv 6333  (class class class)co 7164  ωcom 7593  2_oc2o 8118   +_o coa 8121   ↑_m cmap 8430   ≈ cen 8545  Fincfn 8548   ⊔ cdju 9393  cardccrd 9430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-2o 8125  df-oadd 8128  df-er 8313  df-map 8432  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-dju 9396  df-card 9434

This theorem is referenced by:  ackbij1lem14  9726