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Theorem winalim2 10694
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 10690 . . . 4 (𝐴 ∈ Inaccw β†’ (cardβ€˜π΄) = 𝐴)
2 winainf 10692 . . . . 5 (𝐴 ∈ Inaccw β†’ Ο‰ βŠ† 𝐴)
3 cardalephex 10088 . . . . 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 480 . 2 ((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯))
7 df-rex 3070 . . 3 (βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯) ↔ βˆƒπ‘₯(π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯)))
8 simprr 770 . . . . . . 7 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ 𝐴 = (β„΅β€˜π‘₯))
98eqcomd 2737 . . . . . 6 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ (β„΅β€˜π‘₯) = 𝐴)
10 simprl 768 . . . . . . . 8 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ π‘₯ ∈ On)
11 onzsl 7838 . . . . . . . 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 766 . . . . . . . . . 10 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ 𝐴 β‰  Ο‰)
14 fveq2 6892 . . . . . . . . . . . . . 14 (π‘₯ = βˆ… β†’ (β„΅β€˜π‘₯) = (β„΅β€˜βˆ…))
15 aleph0 10064 . . . . . . . . . . . . . 14 (β„΅β€˜βˆ…) = Ο‰
1614, 15eqtrdi 2787 . . . . . . . . . . . . 13 (π‘₯ = βˆ… β†’ (β„΅β€˜π‘₯) = Ο‰)
17 eqtr 2754 . . . . . . . . . . . . 13 ((𝐴 = (β„΅β€˜π‘₯) ∧ (β„΅β€˜π‘₯) = Ο‰) β†’ 𝐴 = Ο‰)
1816, 17sylan2 592 . . . . . . . . . . . 12 ((𝐴 = (β„΅β€˜π‘₯) ∧ π‘₯ = βˆ…) β†’ 𝐴 = Ο‰)
1918ex 412 . . . . . . . . . . 11 (𝐴 = (β„΅β€˜π‘₯) β†’ (π‘₯ = βˆ… β†’ 𝐴 = Ο‰))
2019necon3ad 2952 . . . . . . . . . 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 5152 . . . . . . . . . . . . . 14 (𝑧 = (β„΅β€˜π‘¦) β†’ (𝑧 β‰Ί 𝑀 ↔ (β„΅β€˜π‘¦) β‰Ί 𝑀))
2423rexbidv 3177 . . . . . . . . . . . . 13 (𝑧 = (β„΅β€˜π‘¦) β†’ (βˆƒπ‘€ ∈ 𝐴 𝑧 β‰Ί 𝑀 ↔ βˆƒπ‘€ ∈ 𝐴 (β„΅β€˜π‘¦) β‰Ί 𝑀))
25 elwina 10684 . . . . . . . . . . . . . . 15 (𝐴 ∈ Inaccw ↔ (𝐴 β‰  βˆ… ∧ (cfβ€˜π΄) = 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 β‰Ί 𝑀))
2625simp3bi 1146 . . . . . . . . . . . . . 14 (𝐴 ∈ Inaccw β†’ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 β‰Ί 𝑀)
2726ad3antrrr 727 . . . . . . . . . . . . 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 7802 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On β†’ suc 𝑦 ∈ On)
29 vex 3477 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
3029sucid 6447 . . . . . . . . . . . . . . . 16 𝑦 ∈ suc 𝑦
31 alephord2i 10075 . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ 𝐴 = (β„΅β€˜π‘₯))
35 fveq2 6892 . . . . . . . . . . . . . . . 16 (π‘₯ = suc 𝑦 β†’ (β„΅β€˜π‘₯) = (β„΅β€˜suc 𝑦))
3635ad2antll 726 . . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . 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 3614 . . . . . . . . . . . 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 456 . . . . . . . . . . 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 9973 . . . . . . . . . . . . . . . . . . 19 ((cardβ€˜π΄) = 𝐴 ↔ (𝐴 ∈ On ∧ βˆ€π‘€ ∈ 𝐴 𝑀 β‰Ί 𝐴))
4241simprbi 496 . . . . . . . . . . . . . . . . . 18 ((cardβ€˜π΄) = 𝐴 β†’ βˆ€π‘€ ∈ 𝐴 𝑀 β‰Ί 𝐴)
43 rsp 3243 . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . 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 5161 . . . . . . . . . . . . . . . 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 10070 . . . . . . . . . . . . . . . 16 Β¬ ((β„΅β€˜π‘¦) β‰Ί 𝑀 ∧ 𝑀 β‰Ί (β„΅β€˜suc 𝑦))
49 pm3.21 471 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . 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 3148 . . . . . . . . . . . 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 456 . . . . . . . . . . 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 3148 . . . . . . . . 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 484 . . . . . . . . 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 1427 . . . . . . 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 511 . . . . 5 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ ((β„΅β€˜π‘₯) = 𝐴 ∧ Lim π‘₯))
6362ex 412 . . . 4 ((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) β†’ ((π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯)) β†’ ((β„΅β€˜π‘₯) = 𝐴 ∧ Lim π‘₯)))
6463eximdv 1919 . . 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 395   ∨ w3o 1085   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069  Vcvv 3473   βŠ† wss 3949  βˆ…c0 4323   class class class wbr 5149  Oncon0 6365  Lim wlim 6366  suc csuc 6367  β€˜cfv 6544  Ο‰com 7858   β‰Ί csdm 8941  cardccrd 9933  β„΅cale 9934  cfccf 9935  Inaccwcwina 10680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-inf2 9639
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7368  df-ov 7415  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-oi 9508  df-har 9555  df-card 9937  df-aleph 9938  df-cf 9939  df-wina 10682
This theorem is referenced by:  winafp  10695
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