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Theorem ackbij1lem5 9301
Description: Lemma for ackbij2 9320. (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
StepHypRef Expression
1 peano2 7286 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2 pw2eng 8275 . . . . . . 7 (suc 𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (2𝑜𝑚 suc 𝐴))
31, 2syl 17 . . . . . 6 (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (2𝑜𝑚 suc 𝐴))
4 df-suc 5916 . . . . . . . . 9 suc 𝐴 = (𝐴 ∪ {𝐴})
54oveq2i 6855 . . . . . . . 8 (2𝑜𝑚 suc 𝐴) = (2𝑜𝑚 (𝐴 ∪ {𝐴}))
6 elex 3365 . . . . . . . . . 10 (𝐴 ∈ ω → 𝐴 ∈ V)
7 snex 5066 . . . . . . . . . . 11 {𝐴} ∈ V
87a1i 11 . . . . . . . . . 10 (𝐴 ∈ ω → {𝐴} ∈ V)
9 2onn 7927 . . . . . . . . . . . 12 2𝑜 ∈ ω
109elexi 3366 . . . . . . . . . . 11 2𝑜 ∈ V
1110a1i 11 . . . . . . . . . 10 (𝐴 ∈ ω → 2𝑜 ∈ V)
12 nnord 7273 . . . . . . . . . . 11 (𝐴 ∈ ω → Ord 𝐴)
13 orddisj 5948 . . . . . . . . . . 11 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
1412, 13syl 17 . . . . . . . . . 10 (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅)
15 mapunen 8338 . . . . . . . . . 10 (((𝐴 ∈ V ∧ {𝐴} ∈ V ∧ 2𝑜 ∈ V) ∧ (𝐴 ∩ {𝐴}) = ∅) → (2𝑜𝑚 (𝐴 ∪ {𝐴})) ≈ ((2𝑜𝑚 𝐴) × (2𝑜𝑚 {𝐴})))
166, 8, 11, 14, 15syl31anc 1492 . . . . . . . . 9 (𝐴 ∈ ω → (2𝑜𝑚 (𝐴 ∪ {𝐴})) ≈ ((2𝑜𝑚 𝐴) × (2𝑜𝑚 {𝐴})))
17 ovex 6876 . . . . . . . . . . . 12 (2𝑜𝑚 𝐴) ∈ V
1817enref 8195 . . . . . . . . . . 11 (2𝑜𝑚 𝐴) ≈ (2𝑜𝑚 𝐴)
1918a1i 11 . . . . . . . . . 10 (𝐴 ∈ ω → (2𝑜𝑚 𝐴) ≈ (2𝑜𝑚 𝐴))
20 2on 7775 . . . . . . . . . . . 12 2𝑜 ∈ On
2120a1i 11 . . . . . . . . . . 11 (𝐴 ∈ ω → 2𝑜 ∈ On)
22 id 22 . . . . . . . . . . 11 (𝐴 ∈ ω → 𝐴 ∈ ω)
2321, 22mapsnend 8241 . . . . . . . . . 10 (𝐴 ∈ ω → (2𝑜𝑚 {𝐴}) ≈ 2𝑜)
24 xpen 8332 . . . . . . . . . 10 (((2𝑜𝑚 𝐴) ≈ (2𝑜𝑚 𝐴) ∧ (2𝑜𝑚 {𝐴}) ≈ 2𝑜) → ((2𝑜𝑚 𝐴) × (2𝑜𝑚 {𝐴})) ≈ ((2𝑜𝑚 𝐴) × 2𝑜))
2519, 23, 24syl2anc 579 . . . . . . . . 9 (𝐴 ∈ ω → ((2𝑜𝑚 𝐴) × (2𝑜𝑚 {𝐴})) ≈ ((2𝑜𝑚 𝐴) × 2𝑜))
26 entr 8214 . . . . . . . . 9 (((2𝑜𝑚 (𝐴 ∪ {𝐴})) ≈ ((2𝑜𝑚 𝐴) × (2𝑜𝑚 {𝐴})) ∧ ((2𝑜𝑚 𝐴) × (2𝑜𝑚 {𝐴})) ≈ ((2𝑜𝑚 𝐴) × 2𝑜)) → (2𝑜𝑚 (𝐴 ∪ {𝐴})) ≈ ((2𝑜𝑚 𝐴) × 2𝑜))
2716, 25, 26syl2anc 579 . . . . . . . 8 (𝐴 ∈ ω → (2𝑜𝑚 (𝐴 ∪ {𝐴})) ≈ ((2𝑜𝑚 𝐴) × 2𝑜))
285, 27syl5eqbr 4846 . . . . . . 7 (𝐴 ∈ ω → (2𝑜𝑚 suc 𝐴) ≈ ((2𝑜𝑚 𝐴) × 2𝑜))
29 pw2eng 8275 . . . . . . . . 9 (𝐴 ∈ ω → 𝒫 𝐴 ≈ (2𝑜𝑚 𝐴))
3010enref 8195 . . . . . . . . . 10 2𝑜 ≈ 2𝑜
3130a1i 11 . . . . . . . . 9 (𝐴 ∈ ω → 2𝑜 ≈ 2𝑜)
32 xpen 8332 . . . . . . . . 9 ((𝒫 𝐴 ≈ (2𝑜𝑚 𝐴) ∧ 2𝑜 ≈ 2𝑜) → (𝒫 𝐴 × 2𝑜) ≈ ((2𝑜𝑚 𝐴) × 2𝑜))
3329, 31, 32syl2anc 579 . . . . . . . 8 (𝐴 ∈ ω → (𝒫 𝐴 × 2𝑜) ≈ ((2𝑜𝑚 𝐴) × 2𝑜))
3433ensymd 8213 . . . . . . 7 (𝐴 ∈ ω → ((2𝑜𝑚 𝐴) × 2𝑜) ≈ (𝒫 𝐴 × 2𝑜))
35 entr 8214 . . . . . . 7 (((2𝑜𝑚 suc 𝐴) ≈ ((2𝑜𝑚 𝐴) × 2𝑜) ∧ ((2𝑜𝑚 𝐴) × 2𝑜) ≈ (𝒫 𝐴 × 2𝑜)) → (2𝑜𝑚 suc 𝐴) ≈ (𝒫 𝐴 × 2𝑜))
3628, 34, 35syl2anc 579 . . . . . 6 (𝐴 ∈ ω → (2𝑜𝑚 suc 𝐴) ≈ (𝒫 𝐴 × 2𝑜))
37 entr 8214 . . . . . 6 ((𝒫 suc 𝐴 ≈ (2𝑜𝑚 suc 𝐴) ∧ (2𝑜𝑚 suc 𝐴) ≈ (𝒫 𝐴 × 2𝑜)) → 𝒫 suc 𝐴 ≈ (𝒫 𝐴 × 2𝑜))
383, 36, 37syl2anc 579 . . . . 5 (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (𝒫 𝐴 × 2𝑜))
39 pwexg 5016 . . . . . 6 (𝐴 ∈ ω → 𝒫 𝐴 ∈ V)
40 xp2cda 9257 . . . . . 6 (𝒫 𝐴 ∈ V → (𝒫 𝐴 × 2𝑜) = (𝒫 𝐴 +𝑐 𝒫 𝐴))
4139, 40syl 17 . . . . 5 (𝐴 ∈ ω → (𝒫 𝐴 × 2𝑜) = (𝒫 𝐴 +𝑐 𝒫 𝐴))
4238, 41breqtrd 4837 . . . 4 (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴))
43 nnfi 8362 . . . . . . . 8 (𝐴 ∈ ω → 𝐴 ∈ Fin)
44 pwfi 8470 . . . . . . . 8 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
4543, 44sylib 209 . . . . . . 7 (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
46 ficardid 9041 . . . . . . 7 (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
4745, 46syl 17 . . . . . 6 (𝐴 ∈ ω → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
48 cdaen 9250 . . . . . 6 (((card‘𝒫 𝐴) ≈ 𝒫 𝐴 ∧ (card‘𝒫 𝐴) ≈ 𝒫 𝐴) → ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)) ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴))
4947, 47, 48syl2anc 579 . . . . 5 (𝐴 ∈ ω → ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)) ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴))
5049ensymd 8213 . . . 4 (𝐴 ∈ ω → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)))
51 entr 8214 . . . 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)))
5242, 50, 51syl2anc 579 . . 3 (𝐴 ∈ ω → 𝒫 suc 𝐴 ≈ ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)))
53 carden2b 9046 . . 3 (𝒫 suc 𝐴 ≈ ((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴)) → (card‘𝒫 suc 𝐴) = (card‘((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴))))
5452, 53syl 17 . 2 (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = (card‘((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴))))
55 ficardom 9040 . . . 4 (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
5645, 55syl 17 . . 3 (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
57 nnacda 9278 . . 3 (((card‘𝒫 𝐴) ∈ ω ∧ (card‘𝒫 𝐴) ∈ ω) → (card‘((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴))) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴)))
5856, 56, 57syl2anc 579 . 2 (𝐴 ∈ ω → (card‘((card‘𝒫 𝐴) +𝑐 (card‘𝒫 𝐴))) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴)))
5954, 58eqtrd 2799 1 (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  Vcvv 3350  cun 3732  cin 3733  c0 4081  𝒫 cpw 4317  {csn 4336   class class class wbr 4811   × cxp 5277  Ord word 5909  Oncon0 5910  suc csuc 5912  cfv 6070  (class class class)co 6844  ωcom 7265  2𝑜c2o 7760   +𝑜 coa 7763  𝑚 cmap 8062  cen 8159  Fincfn 8162  cardccrd 9014   +𝑐 ccda 9244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-2o 7767  df-oadd 7770  df-er 7949  df-map 8064  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-card 9018  df-cda 9245
This theorem is referenced by:  ackbij1lem14  9310
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