MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  winalim2 Structured version   Visualization version   GIF version

Theorem winalim2 10639
Description: A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)
Assertion
Ref Expression
winalim2 ((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) β†’ βˆƒπ‘₯((β„΅β€˜π‘₯) = 𝐴 ∧ Lim π‘₯))
Distinct variable group:   π‘₯,𝐴

Proof of Theorem winalim2
Dummy variables 𝑀 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 winacard 10635 . . . 4 (𝐴 ∈ Inaccw β†’ (cardβ€˜π΄) = 𝐴)
2 winainf 10637 . . . . 5 (𝐴 ∈ Inaccw β†’ Ο‰ βŠ† 𝐴)
3 cardalephex 10033 . . . . 5 (Ο‰ βŠ† 𝐴 β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯)))
42, 3syl 17 . . . 4 (𝐴 ∈ Inaccw β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯)))
51, 4mpbid 231 . . 3 (𝐴 ∈ Inaccw β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯))
65adantr 482 . 2 ((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯))
7 df-rex 3075 . . 3 (βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯) ↔ βˆƒπ‘₯(π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯)))
8 simprr 772 . . . . . . 7 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ 𝐴 = (β„΅β€˜π‘₯))
98eqcomd 2743 . . . . . 6 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ (β„΅β€˜π‘₯) = 𝐴)
10 simprl 770 . . . . . . . 8 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ π‘₯ ∈ On)
11 onzsl 7787 . . . . . . . 8 (π‘₯ ∈ On ↔ (π‘₯ = βˆ… ∨ βˆƒπ‘¦ ∈ On π‘₯ = suc 𝑦 ∨ (π‘₯ ∈ V ∧ Lim π‘₯)))
1210, 11sylib 217 . . . . . . 7 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ (π‘₯ = βˆ… ∨ βˆƒπ‘¦ ∈ On π‘₯ = suc 𝑦 ∨ (π‘₯ ∈ V ∧ Lim π‘₯)))
13 simplr 768 . . . . . . . . . 10 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ 𝐴 β‰  Ο‰)
14 fveq2 6847 . . . . . . . . . . . . . 14 (π‘₯ = βˆ… β†’ (β„΅β€˜π‘₯) = (β„΅β€˜βˆ…))
15 aleph0 10009 . . . . . . . . . . . . . 14 (β„΅β€˜βˆ…) = Ο‰
1614, 15eqtrdi 2793 . . . . . . . . . . . . 13 (π‘₯ = βˆ… β†’ (β„΅β€˜π‘₯) = Ο‰)
17 eqtr 2760 . . . . . . . . . . . . 13 ((𝐴 = (β„΅β€˜π‘₯) ∧ (β„΅β€˜π‘₯) = Ο‰) β†’ 𝐴 = Ο‰)
1816, 17sylan2 594 . . . . . . . . . . . 12 ((𝐴 = (β„΅β€˜π‘₯) ∧ π‘₯ = βˆ…) β†’ 𝐴 = Ο‰)
1918ex 414 . . . . . . . . . . 11 (𝐴 = (β„΅β€˜π‘₯) β†’ (π‘₯ = βˆ… β†’ 𝐴 = Ο‰))
2019necon3ad 2957 . . . . . . . . . 10 (𝐴 = (β„΅β€˜π‘₯) β†’ (𝐴 β‰  Ο‰ β†’ Β¬ π‘₯ = βˆ…))
218, 13, 20sylc 65 . . . . . . . . 9 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ Β¬ π‘₯ = βˆ…)
2221pm2.21d 121 . . . . . . . 8 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ (π‘₯ = βˆ… β†’ Lim π‘₯))
23 breq1 5113 . . . . . . . . . . . . . 14 (𝑧 = (β„΅β€˜π‘¦) β†’ (𝑧 β‰Ί 𝑀 ↔ (β„΅β€˜π‘¦) β‰Ί 𝑀))
2423rexbidv 3176 . . . . . . . . . . . . 13 (𝑧 = (β„΅β€˜π‘¦) β†’ (βˆƒπ‘€ ∈ 𝐴 𝑧 β‰Ί 𝑀 ↔ βˆƒπ‘€ ∈ 𝐴 (β„΅β€˜π‘¦) β‰Ί 𝑀))
25 elwina 10629 . . . . . . . . . . . . . . 15 (𝐴 ∈ Inaccw ↔ (𝐴 β‰  βˆ… ∧ (cfβ€˜π΄) = 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 β‰Ί 𝑀))
2625simp3bi 1148 . . . . . . . . . . . . . 14 (𝐴 ∈ Inaccw β†’ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 β‰Ί 𝑀)
2726ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 β‰Ί 𝑀)
28 onsuc 7751 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On β†’ suc 𝑦 ∈ On)
29 vex 3452 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
3029sucid 6404 . . . . . . . . . . . . . . . 16 𝑦 ∈ suc 𝑦
31 alephord2i 10020 . . . . . . . . . . . . . . . 16 (suc 𝑦 ∈ On β†’ (𝑦 ∈ suc 𝑦 β†’ (β„΅β€˜π‘¦) ∈ (β„΅β€˜suc 𝑦)))
3228, 30, 31mpisyl 21 . . . . . . . . . . . . . . 15 (𝑦 ∈ On β†’ (β„΅β€˜π‘¦) ∈ (β„΅β€˜suc 𝑦))
3332ad2antrl 727 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ (β„΅β€˜π‘¦) ∈ (β„΅β€˜suc 𝑦))
34 simplrr 777 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ 𝐴 = (β„΅β€˜π‘₯))
35 fveq2 6847 . . . . . . . . . . . . . . . 16 (π‘₯ = suc 𝑦 β†’ (β„΅β€˜π‘₯) = (β„΅β€˜suc 𝑦))
3635ad2antll 728 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ (β„΅β€˜π‘₯) = (β„΅β€˜suc 𝑦))
3734, 36eqtrd 2777 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ 𝐴 = (β„΅β€˜suc 𝑦))
3833, 37eleqtrrd 2841 . . . . . . . . . . . . 13 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ (β„΅β€˜π‘¦) ∈ 𝐴)
3924, 27, 38rspcdva 3585 . . . . . . . . . . . 12 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ βˆƒπ‘€ ∈ 𝐴 (β„΅β€˜π‘¦) β‰Ί 𝑀)
4039expr 458 . . . . . . . . . . 11 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ 𝑦 ∈ On) β†’ (π‘₯ = suc 𝑦 β†’ βˆƒπ‘€ ∈ 𝐴 (β„΅β€˜π‘¦) β‰Ί 𝑀))
41 iscard 9918 . . . . . . . . . . . . . . . . . . 19 ((cardβ€˜π΄) = 𝐴 ↔ (𝐴 ∈ On ∧ βˆ€π‘€ ∈ 𝐴 𝑀 β‰Ί 𝐴))
4241simprbi 498 . . . . . . . . . . . . . . . . . 18 ((cardβ€˜π΄) = 𝐴 β†’ βˆ€π‘€ ∈ 𝐴 𝑀 β‰Ί 𝐴)
43 rsp 3233 . . . . . . . . . . . . . . . . . 18 (βˆ€π‘€ ∈ 𝐴 𝑀 β‰Ί 𝐴 β†’ (𝑀 ∈ 𝐴 β†’ 𝑀 β‰Ί 𝐴))
441, 42, 433syl 18 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ Inaccw β†’ (𝑀 ∈ 𝐴 β†’ 𝑀 β‰Ί 𝐴))
4544ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ (𝑀 ∈ 𝐴 β†’ 𝑀 β‰Ί 𝐴))
4637breq2d 5122 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ (𝑀 β‰Ί 𝐴 ↔ 𝑀 β‰Ί (β„΅β€˜suc 𝑦)))
4745, 46sylibd 238 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ (𝑀 ∈ 𝐴 β†’ 𝑀 β‰Ί (β„΅β€˜suc 𝑦)))
48 alephnbtwn2 10015 . . . . . . . . . . . . . . . 16 Β¬ ((β„΅β€˜π‘¦) β‰Ί 𝑀 ∧ 𝑀 β‰Ί (β„΅β€˜suc 𝑦))
49 pm3.21 473 . . . . . . . . . . . . . . . 16 (𝑀 β‰Ί (β„΅β€˜suc 𝑦) β†’ ((β„΅β€˜π‘¦) β‰Ί 𝑀 β†’ ((β„΅β€˜π‘¦) β‰Ί 𝑀 ∧ 𝑀 β‰Ί (β„΅β€˜suc 𝑦))))
5048, 49mtoi 198 . . . . . . . . . . . . . . 15 (𝑀 β‰Ί (β„΅β€˜suc 𝑦) β†’ Β¬ (β„΅β€˜π‘¦) β‰Ί 𝑀)
5147, 50syl6 35 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ (𝑀 ∈ 𝐴 β†’ Β¬ (β„΅β€˜π‘¦) β‰Ί 𝑀))
5251imp 408 . . . . . . . . . . . . 13 (((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) ∧ 𝑀 ∈ 𝐴) β†’ Β¬ (β„΅β€˜π‘¦) β‰Ί 𝑀)
5352nrexdv 3147 . . . . . . . . . . . 12 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ Β¬ βˆƒπ‘€ ∈ 𝐴 (β„΅β€˜π‘¦) β‰Ί 𝑀)
5453expr 458 . . . . . . . . . . 11 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ 𝑦 ∈ On) β†’ (π‘₯ = suc 𝑦 β†’ Β¬ βˆƒπ‘€ ∈ 𝐴 (β„΅β€˜π‘¦) β‰Ί 𝑀))
5540, 54pm2.65d 195 . . . . . . . . . 10 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ 𝑦 ∈ On) β†’ Β¬ π‘₯ = suc 𝑦)
5655nrexdv 3147 . . . . . . . . 9 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ Β¬ βˆƒπ‘¦ ∈ On π‘₯ = suc 𝑦)
5756pm2.21d 121 . . . . . . . 8 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ (βˆƒπ‘¦ ∈ On π‘₯ = suc 𝑦 β†’ Lim π‘₯))
58 simpr 486 . . . . . . . . 9 ((π‘₯ ∈ V ∧ Lim π‘₯) β†’ Lim π‘₯)
5958a1i 11 . . . . . . . 8 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ ((π‘₯ ∈ V ∧ Lim π‘₯) β†’ Lim π‘₯))
6022, 57, 593jaod 1429 . . . . . . 7 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ ((π‘₯ = βˆ… ∨ βˆƒπ‘¦ ∈ On π‘₯ = suc 𝑦 ∨ (π‘₯ ∈ V ∧ Lim π‘₯)) β†’ Lim π‘₯))
6112, 60mpd 15 . . . . . 6 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ Lim π‘₯)
629, 61jca 513 . . . . 5 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ ((β„΅β€˜π‘₯) = 𝐴 ∧ Lim π‘₯))
6362ex 414 . . . 4 ((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) β†’ ((π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯)) β†’ ((β„΅β€˜π‘₯) = 𝐴 ∧ Lim π‘₯)))
6463eximdv 1921 . . 3 ((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) β†’ (βˆƒπ‘₯(π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯)) β†’ βˆƒπ‘₯((β„΅β€˜π‘₯) = 𝐴 ∧ Lim π‘₯)))
657, 64biimtrid 241 . 2 ((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) β†’ (βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯) β†’ βˆƒπ‘₯((β„΅β€˜π‘₯) = 𝐴 ∧ Lim π‘₯)))
666, 65mpd 15 1 ((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) β†’ βˆƒπ‘₯((β„΅β€˜π‘₯) = 𝐴 ∧ Lim π‘₯))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ w3o 1087   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287   class class class wbr 5110  Oncon0 6322  Lim wlim 6323  suc csuc 6324  β€˜cfv 6501  Ο‰com 7807   β‰Ί csdm 8889  cardccrd 9878  β„΅cale 9879  cfccf 9880  Inaccwcwina 10625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9453  df-har 9500  df-card 9882  df-aleph 9883  df-cf 9884  df-wina 10627
This theorem is referenced by:  winafp  10640
  Copyright terms: Public domain W3C validator