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Theorem fineqvlem 8423
Description: Lemma for fineqv 8424. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
fineqvlem ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴)

Proof of Theorem fineqvlem
Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 5061 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
21adantr 468 . . 3 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → 𝒫 𝐴 ∈ V)
32pwexd 5062 . 2 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → 𝒫 𝒫 𝐴 ∈ V)
4 ssrab2 3895 . . . . 5 {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ⊆ 𝒫 𝐴
5 elpw2g 5032 . . . . . 6 (𝒫 𝐴 ∈ V → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ⊆ 𝒫 𝐴))
62, 5syl 17 . . . . 5 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ⊆ 𝒫 𝐴))
74, 6mpbiri 249 . . . 4 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ∈ 𝒫 𝒫 𝐴)
87a1d 25 . . 3 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (𝑏 ∈ ω → {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ∈ 𝒫 𝒫 𝐴))
9 isinf 8422 . . . . . . . . 9 𝐴 ∈ Fin → ∀𝑏 ∈ ω ∃𝑒(𝑒𝐴𝑒𝑏))
109r19.21bi 3131 . . . . . . . 8 ((¬ 𝐴 ∈ Fin ∧ 𝑏 ∈ ω) → ∃𝑒(𝑒𝐴𝑒𝑏))
1110ad2ant2lr 745 . . . . . . 7 (((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ∃𝑒(𝑒𝐴𝑒𝑏))
12 selpw 4369 . . . . . . . . . . 11 (𝑒 ∈ 𝒫 𝐴𝑒𝐴)
1312biimpri 219 . . . . . . . . . 10 (𝑒𝐴𝑒 ∈ 𝒫 𝐴)
1413anim1i 604 . . . . . . . . 9 ((𝑒𝐴𝑒𝑏) → (𝑒 ∈ 𝒫 𝐴𝑒𝑏))
15 breq1 4858 . . . . . . . . . 10 (𝑑 = 𝑒 → (𝑑𝑏𝑒𝑏))
1615elrab 3570 . . . . . . . . 9 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ↔ (𝑒 ∈ 𝒫 𝐴𝑒𝑏))
1714, 16sylibr 225 . . . . . . . 8 ((𝑒𝐴𝑒𝑏) → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏})
1817eximi 1919 . . . . . . 7 (∃𝑒(𝑒𝐴𝑒𝑏) → ∃𝑒 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏})
1911, 18syl 17 . . . . . 6 (((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ∃𝑒 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏})
20 eleq2 2885 . . . . . . . . 9 ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ↔ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐}))
2120biimpcd 240 . . . . . . . 8 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐}))
2221adantl 469 . . . . . . 7 ((((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏}) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐}))
2316simprbi 486 . . . . . . . . . 10 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} → 𝑒𝑏)
24 breq1 4858 . . . . . . . . . . . 12 (𝑑 = 𝑒 → (𝑑𝑐𝑒𝑐))
2524elrab 3570 . . . . . . . . . . 11 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐} ↔ (𝑒 ∈ 𝒫 𝐴𝑒𝑐))
2625simprbi 486 . . . . . . . . . 10 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑒𝑐)
27 ensym 8251 . . . . . . . . . . 11 (𝑒𝑏𝑏𝑒)
28 entr 8254 . . . . . . . . . . 11 ((𝑏𝑒𝑒𝑐) → 𝑏𝑐)
2927, 28sylan 571 . . . . . . . . . 10 ((𝑒𝑏𝑒𝑐) → 𝑏𝑐)
3023, 26, 29syl2an 585 . . . . . . . . 9 ((𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐}) → 𝑏𝑐)
3130ex 399 . . . . . . . 8 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑏𝑐))
3231adantl 469 . . . . . . 7 ((((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏}) → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑏𝑐))
33 nneneq 8392 . . . . . . . . 9 ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏𝑐𝑏 = 𝑐))
3433biimpd 220 . . . . . . . 8 ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏𝑐𝑏 = 𝑐))
3534ad2antlr 709 . . . . . . 7 ((((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏}) → (𝑏𝑐𝑏 = 𝑐))
3622, 32, 353syld 60 . . . . . 6 ((((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏}) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑏 = 𝑐))
3719, 36exlimddv 2026 . . . . 5 (((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑏 = 𝑐))
38 breq2 4859 . . . . . 6 (𝑏 = 𝑐 → (𝑑𝑏𝑑𝑐))
3938rabbidv 3390 . . . . 5 (𝑏 = 𝑐 → {𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐})
4037, 39impbid1 216 . . . 4 (((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} ↔ 𝑏 = 𝑐))
4140ex 399 . . 3 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} ↔ 𝑏 = 𝑐)))
428, 41dom2d 8243 . 2 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (𝒫 𝒫 𝐴 ∈ V → ω ≼ 𝒫 𝒫 𝐴))
433, 42mpd 15 1 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  wex 1859  wcel 2157  {crab 3111  Vcvv 3402  wss 3780  𝒫 cpw 4362   class class class wbr 4855  ωcom 7305  cen 8199  cdom 8200  Fincfn 8202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-om 7306  df-er 7989  df-en 8203  df-dom 8204  df-fin 8206
This theorem is referenced by:  fineqv  8424  isfin1-2  9502
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