Theorem ackbij1lem5 9046

Description: Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Assertion

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

Proof of Theorem ackbij1lem5

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suceq 5790	. . . . 5 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
2	1	pweqd 4163	. . . 4 ⊢ (𝑎 = 𝐴 → 𝒫 suc 𝑎 = 𝒫 suc 𝐴)
3	2	fveq2d 6195	. . 3 ⊢ (𝑎 = 𝐴 → (card‘𝒫 suc 𝑎) = (card‘𝒫 suc 𝐴))
4		pweq 4161	. . . . 5 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
5	4	fveq2d 6195	. . . 4 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
6	5, 5	oveq12d 6668	. . 3 ⊢ (𝑎 = 𝐴 → ((card‘𝒫 𝑎) +𝑜 (card‘𝒫 𝑎)) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴)))
7	3, 6	eqeq12d 2637	. 2 ⊢ (𝑎 = 𝐴 → ((card‘𝒫 suc 𝑎) = ((card‘𝒫 𝑎) +𝑜 (card‘𝒫 𝑎)) ↔ (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴))))
8		vex 3203	. . . . . . . . 9 ⊢ 𝑎 ∈ V
9	8	sucex 7011	. . . . . . . 8 ⊢ suc 𝑎 ∈ V
10	9	pw2en 8067	. . . . . . 7 ⊢ 𝒫 suc 𝑎 ≈ (2𝑜 ↑𝑚 suc 𝑎)
11		df-suc 5729	. . . . . . . . . 10 ⊢ suc 𝑎 = (𝑎 ∪ {𝑎})
12	11	oveq2i 6661	. . . . . . . . 9 ⊢ (2𝑜 ↑𝑚 suc 𝑎) = (2𝑜 ↑𝑚 (𝑎 ∪ {𝑎}))
13		nnord 7073	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ω → Ord 𝑎)
14		orddisj 5762	. . . . . . . . . . 11 ⊢ (Ord 𝑎 → (𝑎 ∩ {𝑎}) = ∅)
15		snex 4908	. . . . . . . . . . . 12 ⊢ {𝑎} ∈ V
16		2onn 7720	. . . . . . . . . . . . 13 ⊢ 2𝑜 ∈ ω
17	16	elexi 3213	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ V
18		mapunen 8129	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ V ∧ {𝑎} ∈ V ∧ 2𝑜 ∈ V) ∧ (𝑎 ∩ {𝑎}) = ∅) → (2𝑜 ↑𝑚 (𝑎 ∪ {𝑎})) ≈ ((2𝑜 ↑𝑚 𝑎) × (2𝑜 ↑𝑚 {𝑎})))
19	18	ex 450	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ V ∧ {𝑎} ∈ V ∧ 2𝑜 ∈ V) → ((𝑎 ∩ {𝑎}) = ∅ → (2𝑜 ↑𝑚 (𝑎 ∪ {𝑎})) ≈ ((2𝑜 ↑𝑚 𝑎) × (2𝑜 ↑𝑚 {𝑎}))))
20	8, 15, 17, 19	mp3an 1424	. . . . . . . . . . 11 ⊢ ((𝑎 ∩ {𝑎}) = ∅ → (2𝑜 ↑𝑚 (𝑎 ∪ {𝑎})) ≈ ((2𝑜 ↑𝑚 𝑎) × (2𝑜 ↑𝑚 {𝑎})))
21	13, 14, 20	3syl 18	. . . . . . . . . 10 ⊢ (𝑎 ∈ ω → (2𝑜 ↑𝑚 (𝑎 ∪ {𝑎})) ≈ ((2𝑜 ↑𝑚 𝑎) × (2𝑜 ↑𝑚 {𝑎})))
22		ovex 6678	. . . . . . . . . . . 12 ⊢ (2𝑜 ↑𝑚 𝑎) ∈ V
23	22	enref 7988	. . . . . . . . . . 11 ⊢ (2𝑜 ↑𝑚 𝑎) ≈ (2𝑜 ↑𝑚 𝑎)
24	17, 8	mapsnen 8035	. . . . . . . . . . 11 ⊢ (2𝑜 ↑𝑚 {𝑎}) ≈ 2𝑜
25		xpen 8123	. . . . . . . . . . 11 ⊢ (((2𝑜 ↑𝑚 𝑎) ≈ (2𝑜 ↑𝑚 𝑎) ∧ (2𝑜 ↑𝑚 {𝑎}) ≈ 2𝑜) → ((2𝑜 ↑𝑚 𝑎) × (2𝑜 ↑𝑚 {𝑎})) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜))
26	23, 24, 25	mp2an 708	. . . . . . . . . 10 ⊢ ((2𝑜 ↑𝑚 𝑎) × (2𝑜 ↑𝑚 {𝑎})) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜)
27		entr 8008	. . . . . . . . . 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))
28	21, 26, 27	sylancl 694	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (2𝑜 ↑𝑚 (𝑎 ∪ {𝑎})) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜))
29	12, 28	syl5eqbr 4688	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (2𝑜 ↑𝑚 suc 𝑎) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜))
30	8	pw2en 8067	. . . . . . . . . 10 ⊢ 𝒫 𝑎 ≈ (2𝑜 ↑𝑚 𝑎)
31	17	enref 7988	. . . . . . . . . 10 ⊢ 2𝑜 ≈ 2𝑜
32		xpen 8123	. . . . . . . . . 10 ⊢ ((𝒫 𝑎 ≈ (2𝑜 ↑𝑚 𝑎) ∧ 2𝑜 ≈ 2𝑜) → (𝒫 𝑎 × 2𝑜) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜))
33	30, 31, 32	mp2an 708	. . . . . . . . 9 ⊢ (𝒫 𝑎 × 2𝑜) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜)
34	33	ensymi 8006	. . . . . . . 8 ⊢ ((2𝑜 ↑𝑚 𝑎) × 2𝑜) ≈ (𝒫 𝑎 × 2𝑜)
35		entr 8008	. . . . . . . 8 ⊢ (((2𝑜 ↑𝑚 suc 𝑎) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜) ∧ ((2𝑜 ↑𝑚 𝑎) × 2𝑜) ≈ (𝒫 𝑎 × 2𝑜)) → (2𝑜 ↑𝑚 suc 𝑎) ≈ (𝒫 𝑎 × 2𝑜))
36	29, 34, 35	sylancl 694	. . . . . . 7 ⊢ (𝑎 ∈ ω → (2𝑜 ↑𝑚 suc 𝑎) ≈ (𝒫 𝑎 × 2𝑜))
37		entr 8008	. . . . . . 7 ⊢ ((𝒫 suc 𝑎 ≈ (2𝑜 ↑𝑚 suc 𝑎) ∧ (2𝑜 ↑𝑚 suc 𝑎) ≈ (𝒫 𝑎 × 2𝑜)) → 𝒫 suc 𝑎 ≈ (𝒫 𝑎 × 2𝑜))
38	10, 36, 37	sylancr 695	. . . . . 6 ⊢ (𝑎 ∈ ω → 𝒫 suc 𝑎 ≈ (𝒫 𝑎 × 2𝑜))
39		vpwex 4849	. . . . . . 7 ⊢ 𝒫 𝑎 ∈ V
40		xp2cda 9002	. . . . . . 7 ⊢ (𝒫 𝑎 ∈ V → (𝒫 𝑎 × 2𝑜) = (𝒫 𝑎 +𝑐 𝒫 𝑎))
41	39, 40	ax-mp 5	. . . . . 6 ⊢ (𝒫 𝑎 × 2𝑜) = (𝒫 𝑎 +𝑐 𝒫 𝑎)
42	38, 41	syl6breq 4694	. . . . 5 ⊢ (𝑎 ∈ ω → 𝒫 suc 𝑎 ≈ (𝒫 𝑎 +𝑐 𝒫 𝑎))
43		nnfi 8153	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
44		pwfi 8261	. . . . . . . . 9 ⊢ (𝑎 ∈ Fin ↔ 𝒫 𝑎 ∈ Fin)
45	43, 44	sylib 208	. . . . . . . 8 ⊢ (𝑎 ∈ ω → 𝒫 𝑎 ∈ Fin)
46		ficardid 8788	. . . . . . . 8 ⊢ (𝒫 𝑎 ∈ Fin → (card‘𝒫 𝑎) ≈ 𝒫 𝑎)
47	45, 46	syl 17	. . . . . . 7 ⊢ (𝑎 ∈ ω → (card‘𝒫 𝑎) ≈ 𝒫 𝑎)
48		cdaen 8995	. . . . . . 7 ⊢ (((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 693	. . . . . 6 ⊢ (𝑎 ∈ ω → ((card‘𝒫 𝑎) +𝑐 (card‘𝒫 𝑎)) ≈ (𝒫 𝑎 +𝑐 𝒫 𝑎))
50	49	ensymd 8007	. . . . 5 ⊢ (𝑎 ∈ ω → (𝒫 𝑎 +𝑐 𝒫 𝑎) ≈ ((card‘𝒫 𝑎) +𝑐 (card‘𝒫 𝑎)))
51		entr 8008	. . . . 5 ⊢ ((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 693	. . . 4 ⊢ (𝑎 ∈ ω → 𝒫 suc 𝑎 ≈ ((card‘𝒫 𝑎) +𝑐 (card‘𝒫 𝑎)))
53		carden2b 8793	. . . 4 ⊢ (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	. . 3 ⊢ (𝑎 ∈ ω → (card‘𝒫 suc 𝑎) = (card‘((card‘𝒫 𝑎) +𝑐 (card‘𝒫 𝑎))))
55		ficardom 8787	. . . . 5 ⊢ (𝒫 𝑎 ∈ Fin → (card‘𝒫 𝑎) ∈ ω)
56	45, 55	syl 17	. . . 4 ⊢ (𝑎 ∈ ω → (card‘𝒫 𝑎) ∈ ω)
57		nnacda 9023	. . . 4 ⊢ (((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 693	. . 3 ⊢ (𝑎 ∈ ω → (card‘((card‘𝒫 𝑎) +𝑐 (card‘𝒫 𝑎))) = ((card‘𝒫 𝑎) +𝑜 (card‘𝒫 𝑎)))
59	54, 58	eqtrd 2656	. 2 ⊢ (𝑎 ∈ ω → (card‘𝒫 suc 𝑎) = ((card‘𝒫 𝑎) +𝑜 (card‘𝒫 𝑎)))
60	7, 59	vtoclga 3272	1 ⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴)))

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  𝒫 cpw 4158  {csn 4177   class class class wbr 4653   × cxp 5112  Ord word 5722  suc csuc 5725  ‘cfv 5888  (class class class)co 6650  ωcom 7065  2_𝑜c2o 7554   +_𝑜 coa 7557   ↑_𝑚 cmap 7857   ≈ cen 7952  Fincfn 7955  cardccrd 8761   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990

This theorem is referenced by:  ackbij1lem14  9055