Theorem ackbij1lem5 8929

Description: Lemma for ackbij2 8948. (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 5707	. . . . 5 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
2	1	pweqd 4113	. . . 4 ⊢ (𝑎 = 𝐴 → 𝒫 suc 𝑎 = 𝒫 suc 𝐴)
3	2	fveq2d 6107	. . 3 ⊢ (𝑎 = 𝐴 → (card‘𝒫 suc 𝑎) = (card‘𝒫 suc 𝐴))
4		pweq 4111	. . . . 5 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
5	4	fveq2d 6107	. . . 4 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
6	5, 5	oveq12d 6567	. . 3 ⊢ (𝑎 = 𝐴 → ((card‘𝒫 𝑎) +𝑜 (card‘𝒫 𝑎)) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴)))
7	3, 6	eqeq12d 2625	. 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 3176	. . . . . . . . 9 ⊢ 𝑎 ∈ V
9	8	sucex 6903	. . . . . . . 8 ⊢ suc 𝑎 ∈ V
10	9	pw2en 7952	. . . . . . 7 ⊢ 𝒫 suc 𝑎 ≈ (2𝑜 ↑𝑚 suc 𝑎)
11		df-suc 5646	. . . . . . . . . 10 ⊢ suc 𝑎 = (𝑎 ∪ {𝑎})
12	11	oveq2i 6560	. . . . . . . . 9 ⊢ (2𝑜 ↑𝑚 suc 𝑎) = (2𝑜 ↑𝑚 (𝑎 ∪ {𝑎}))
13		nnord 6965	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ω → Ord 𝑎)
14		orddisj 5679	. . . . . . . . . . 11 ⊢ (Ord 𝑎 → (𝑎 ∩ {𝑎}) = ∅)
15		snex 4835	. . . . . . . . . . . 12 ⊢ {𝑎} ∈ V
16		2onn 7607	. . . . . . . . . . . . 13 ⊢ 2𝑜 ∈ ω
17	16	elexi 3186	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ V
18		mapunen 8014	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ V ∧ {𝑎} ∈ V ∧ 2𝑜 ∈ V) ∧ (𝑎 ∩ {𝑎}) = ∅) → (2𝑜 ↑𝑚 (𝑎 ∪ {𝑎})) ≈ ((2𝑜 ↑𝑚 𝑎) × (2𝑜 ↑𝑚 {𝑎})))
19	18	ex 449	. . . . . . . . . . . 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 1416	. . . . . . . . . . 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 6577	. . . . . . . . . . . 12 ⊢ (2𝑜 ↑𝑚 𝑎) ∈ V
23	22	enref 7874	. . . . . . . . . . 11 ⊢ (2𝑜 ↑𝑚 𝑎) ≈ (2𝑜 ↑𝑚 𝑎)
24	17, 8	mapsnen 7920	. . . . . . . . . . 11 ⊢ (2𝑜 ↑𝑚 {𝑎}) ≈ 2𝑜
25		xpen 8008	. . . . . . . . . . 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 704	. . . . . . . . . 10 ⊢ ((2𝑜 ↑𝑚 𝑎) × (2𝑜 ↑𝑚 {𝑎})) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜)
27		entr 7894	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (2𝑜 ↑𝑚 (𝑎 ∪ {𝑎})) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜))
29	12, 28	syl5eqbr 4618	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (2𝑜 ↑𝑚 suc 𝑎) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜))
30	8	pw2en 7952	. . . . . . . . . 10 ⊢ 𝒫 𝑎 ≈ (2𝑜 ↑𝑚 𝑎)
31	17	enref 7874	. . . . . . . . . 10 ⊢ 2𝑜 ≈ 2𝑜
32		xpen 8008	. . . . . . . . . 10 ⊢ ((𝒫 𝑎 ≈ (2𝑜 ↑𝑚 𝑎) ∧ 2𝑜 ≈ 2𝑜) → (𝒫 𝑎 × 2𝑜) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜))
33	30, 31, 32	mp2an 704	. . . . . . . . 9 ⊢ (𝒫 𝑎 × 2𝑜) ≈ ((2𝑜 ↑𝑚 𝑎) × 2𝑜)
34	33	ensymi 7892	. . . . . . . 8 ⊢ ((2𝑜 ↑𝑚 𝑎) × 2𝑜) ≈ (𝒫 𝑎 × 2𝑜)
35		entr 7894	. . . . . . . 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 693	. . . . . . 7 ⊢ (𝑎 ∈ ω → (2𝑜 ↑𝑚 suc 𝑎) ≈ (𝒫 𝑎 × 2𝑜))
37		entr 7894	. . . . . . 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 694	. . . . . 6 ⊢ (𝑎 ∈ ω → 𝒫 suc 𝑎 ≈ (𝒫 𝑎 × 2𝑜))
39		vpwex 4775	. . . . . . 7 ⊢ 𝒫 𝑎 ∈ V
40		xp2cda 8885	. . . . . . 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 4624	. . . . 5 ⊢ (𝑎 ∈ ω → 𝒫 suc 𝑎 ≈ (𝒫 𝑎 +𝑐 𝒫 𝑎))
43		nnfi 8038	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
44		pwfi 8144	. . . . . . . . 9 ⊢ (𝑎 ∈ Fin ↔ 𝒫 𝑎 ∈ Fin)
45	43, 44	sylib 207	. . . . . . . 8 ⊢ (𝑎 ∈ ω → 𝒫 𝑎 ∈ Fin)
46		ficardid 8671	. . . . . . . 8 ⊢ (𝒫 𝑎 ∈ Fin → (card‘𝒫 𝑎) ≈ 𝒫 𝑎)
47	45, 46	syl 17	. . . . . . 7 ⊢ (𝑎 ∈ ω → (card‘𝒫 𝑎) ≈ 𝒫 𝑎)
48		cdaen 8878	. . . . . . 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 691	. . . . . 6 ⊢ (𝑎 ∈ ω → ((card‘𝒫 𝑎) +𝑐 (card‘𝒫 𝑎)) ≈ (𝒫 𝑎 +𝑐 𝒫 𝑎))
50	49	ensymd 7893	. . . . 5 ⊢ (𝑎 ∈ ω → (𝒫 𝑎 +𝑐 𝒫 𝑎) ≈ ((card‘𝒫 𝑎) +𝑐 (card‘𝒫 𝑎)))
51		entr 7894	. . . . 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 691	. . . 4 ⊢ (𝑎 ∈ ω → 𝒫 suc 𝑎 ≈ ((card‘𝒫 𝑎) +𝑐 (card‘𝒫 𝑎)))
53		carden2b 8676	. . . 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 8670	. . . . 5 ⊢ (𝒫 𝑎 ∈ Fin → (card‘𝒫 𝑎) ∈ ω)
56	45, 55	syl 17	. . . 4 ⊢ (𝑎 ∈ ω → (card‘𝒫 𝑎) ∈ ω)
57		nnacda 8906	. . . 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 691	. . 3 ⊢ (𝑎 ∈ ω → (card‘((card‘𝒫 𝑎) +𝑐 (card‘𝒫 𝑎))) = ((card‘𝒫 𝑎) +𝑜 (card‘𝒫 𝑎)))
59	54, 58	eqtrd 2644	. 2 ⊢ (𝑎 ∈ ω → (card‘𝒫 suc 𝑎) = ((card‘𝒫 𝑎) +𝑜 (card‘𝒫 𝑎)))
60	7, 59	vtoclga 3245	1 ⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴)))

Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583   × cxp 5036  Ord word 5639  suc csuc 5642  ‘cfv 5804  (class class class)co 6549  ωcom 6957  2_𝑜c2o 7441   +_𝑜 coa 7444   ↑_𝑚 cmap 7744   ≈ cen 7838  Fincfn 7841  cardccrd 8644   +_𝑐 ccda 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873

This theorem is referenced by:  ackbij1lem14  8938