Theorem ackbij1lem5 10145

Description: Lemma for ackbij2 10164. (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 7842	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2		pw2eng 9023	. . . . . . 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 6331	. . . . . . . . . 10 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5	4	oveq2i 7379	. . . . . . . . 9 ⊢ (2o ↑m suc 𝐴) = (2o ↑m (𝐴 ∪ {𝐴}))
6		elex 3463	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → 𝐴 ∈ V)
7		snex 5385	. . . . . . . . . . . 12 ⊢ {𝐴} ∈ V
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → {𝐴} ∈ V)
9		2onn 8580	. . . . . . . . . . . . 13 ⊢ 2o ∈ ω
10	9	elexi 3465	. . . . . . . . . . . 12 ⊢ 2o ∈ V
11	10	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → 2o ∈ V)
12		nnord 7826	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → Ord 𝐴)
13		orddisj 6363	. . . . . . . . . . . 12 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅)
15		mapunen 9086	. . . . . . . . . . 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 1376	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (2o ↑m (𝐴 ∪ {𝐴})) ≈ ((2o ↑m 𝐴) × (2o ↑m {𝐴})))
17		ovex 7401	. . . . . . . . . . . 12 ⊢ (2o ↑m 𝐴) ∈ V
18	17	enref 8934	. . . . . . . . . . 11 ⊢ (2o ↑m 𝐴) ≈ (2o ↑m 𝐴)
19		2on 8420	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 2o ∈ On)
21		id 22	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 𝐴 ∈ ω)
22	20, 21	mapsnend 8985	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (2o ↑m {𝐴}) ≈ 2o)
23		xpen 9080	. . . . . . . . . . 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 588	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → ((2o ↑m 𝐴) × (2o ↑m {𝐴})) ≈ ((2o ↑m 𝐴) × 2o))
25		entr 8955	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (2o ↑m (𝐴 ∪ {𝐴})) ≈ ((2o ↑m 𝐴) × 2o))
27	5, 26	eqbrtrid 5135	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (2o ↑m suc 𝐴) ≈ ((2o ↑m 𝐴) × 2o))
28	17, 10	xpcomen 9008	. . . . . . . 8 ⊢ ((2o ↑m 𝐴) × 2o) ≈ (2o × (2o ↑m 𝐴))
29		entr 8955	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝐴 ∈ ω → (2o ↑m suc 𝐴) ≈ (2o × (2o ↑m 𝐴)))
31	10	enref 8934	. . . . . . . . 9 ⊢ 2o ≈ 2o
32		pw2eng 9023	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ≈ (2o ↑m 𝐴))
33		xpen 9080	. . . . . . . . 9 ⊢ ((2o ≈ 2o ∧ 𝒫 𝐴 ≈ (2o ↑m 𝐴)) → (2o × 𝒫 𝐴) ≈ (2o × (2o ↑m 𝐴)))
34	31, 32, 33	sylancr 588	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (2o × 𝒫 𝐴) ≈ (2o × (2o ↑m 𝐴)))
35	34	ensymd 8954	. . . . . . 7 ⊢ (𝐴 ∈ ω → (2o × (2o ↑m 𝐴)) ≈ (2o × 𝒫 𝐴))
36		entr 8955	. . . . . . 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 585	. . . . . 6 ⊢ (𝐴 ∈ ω → (2o ↑m suc 𝐴) ≈ (2o × 𝒫 𝐴))
38		entr 8955	. . . . . 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 585	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (2o × 𝒫 𝐴))
40		xp2dju 10099	. . . . 5 ⊢ (2o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴)
41	39, 40	breqtrdi 5141	. . . 4 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴))
42		nnfi 9104	. . . . . . . 8 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
43		pwfi 9231	. . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
44	42, 43	sylib 218	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
45		ficardid 9886	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
46	44, 45	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
47		djuen 10092	. . . . . 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 585	. . . . 5 ⊢ (𝐴 ∈ ω → ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)) ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴))
49	48	ensymd 8954	. . . 4 ⊢ (𝐴 ∈ ω → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)))
50		entr 8955	. . . 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 585	. . 3 ⊢ (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ ((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴)))
52		carden2b 9891	. . 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 9885	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
55	44, 54	syl 17	. . 3 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
56		nnadju 10120	. . 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 585	. 2 ⊢ (𝐴 ∈ ω → (card‘((card‘𝒫 𝐴) ⊔ (card‘𝒫 𝐴))) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))
58	53, 57	eqtrd 2772	1 ⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   ∩ cin 3902  ∅c0 4287  𝒫 cpw 4556  {csn 4582   class class class wbr 5100   × cxp 5630  Ord word 6324  Oncon0 6325  suc csuc 6327  ‘cfv 6500  (class class class)co 7368  ωcom 7818  2_oc2o 8401   +_o coa 8404   ↑_m cmap 8775   ≈ cen 8892  Fincfn 8895   ⊔ cdju 9822  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863

This theorem is referenced by:  ackbij1lem14  10154