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Theorem winalim2 10693
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 10689 . . . 4 (𝐴 ∈ Inaccw β†’ (cardβ€˜π΄) = 𝐴)
2 winainf 10691 . . . . 5 (𝐴 ∈ Inaccw β†’ Ο‰ βŠ† 𝐴)
3 cardalephex 10087 . . . . 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 479 . 2 ((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯))
7 df-rex 3069 . . 3 (βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯) ↔ βˆƒπ‘₯(π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯)))
8 simprr 769 . . . . . . 7 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ 𝐴 = (β„΅β€˜π‘₯))
98eqcomd 2736 . . . . . 6 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ (β„΅β€˜π‘₯) = 𝐴)
10 simprl 767 . . . . . . . 8 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ π‘₯ ∈ On)
11 onzsl 7837 . . . . . . . 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 765 . . . . . . . . . 10 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ 𝐴 β‰  Ο‰)
14 fveq2 6890 . . . . . . . . . . . . . 14 (π‘₯ = βˆ… β†’ (β„΅β€˜π‘₯) = (β„΅β€˜βˆ…))
15 aleph0 10063 . . . . . . . . . . . . . 14 (β„΅β€˜βˆ…) = Ο‰
1614, 15eqtrdi 2786 . . . . . . . . . . . . 13 (π‘₯ = βˆ… β†’ (β„΅β€˜π‘₯) = Ο‰)
17 eqtr 2753 . . . . . . . . . . . . 13 ((𝐴 = (β„΅β€˜π‘₯) ∧ (β„΅β€˜π‘₯) = Ο‰) β†’ 𝐴 = Ο‰)
1816, 17sylan2 591 . . . . . . . . . . . 12 ((𝐴 = (β„΅β€˜π‘₯) ∧ π‘₯ = βˆ…) β†’ 𝐴 = Ο‰)
1918ex 411 . . . . . . . . . . 11 (𝐴 = (β„΅β€˜π‘₯) β†’ (π‘₯ = βˆ… β†’ 𝐴 = Ο‰))
2019necon3ad 2951 . . . . . . . . . 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 5150 . . . . . . . . . . . . . 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 10683 . . . . . . . . . . . . . . 15 (𝐴 ∈ Inaccw ↔ (𝐴 β‰  βˆ… ∧ (cfβ€˜π΄) = 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 β‰Ί 𝑀))
2625simp3bi 1145 . . . . . . . . . . . . . 14 (𝐴 ∈ Inaccw β†’ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 β‰Ί 𝑀)
2726ad3antrrr 726 . . . . . . . . . . . . 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 7801 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On β†’ suc 𝑦 ∈ On)
29 vex 3476 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
3029sucid 6445 . . . . . . . . . . . . . . . 16 𝑦 ∈ suc 𝑦
31 alephord2i 10074 . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) ∧ (𝑦 ∈ On ∧ π‘₯ = suc 𝑦)) β†’ 𝐴 = (β„΅β€˜π‘₯))
35 fveq2 6890 . . . . . . . . . . . . . . . 16 (π‘₯ = suc 𝑦 β†’ (β„΅β€˜π‘₯) = (β„΅β€˜suc 𝑦))
3635ad2antll 725 . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . 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 3612 . . . . . . . . . . . 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 455 . . . . . . . . . . 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 9972 . . . . . . . . . . . . . . . . . . 19 ((cardβ€˜π΄) = 𝐴 ↔ (𝐴 ∈ On ∧ βˆ€π‘€ ∈ 𝐴 𝑀 β‰Ί 𝐴))
4241simprbi 495 . . . . . . . . . . . . . . . . . 18 ((cardβ€˜π΄) = 𝐴 β†’ βˆ€π‘€ ∈ 𝐴 𝑀 β‰Ί 𝐴)
43 rsp 3242 . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 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 5159 . . . . . . . . . . . . . . . 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 10069 . . . . . . . . . . . . . . . 16 Β¬ ((β„΅β€˜π‘¦) β‰Ί 𝑀 ∧ 𝑀 β‰Ί (β„΅β€˜suc 𝑦))
49 pm3.21 470 . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . 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 455 . . . . . . . . . . 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 483 . . . . . . . . 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 1426 . . . . . . 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 510 . . . . 5 (((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) ∧ (π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯))) β†’ ((β„΅β€˜π‘₯) = 𝐴 ∧ Lim π‘₯))
6362ex 411 . . . 4 ((𝐴 ∈ Inaccw ∧ 𝐴 β‰  Ο‰) β†’ ((π‘₯ ∈ On ∧ 𝐴 = (β„΅β€˜π‘₯)) β†’ ((β„΅β€˜π‘₯) = 𝐴 ∧ Lim π‘₯)))
6463eximdv 1918 . . 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 394   ∨ w3o 1084   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βŠ† wss 3947  βˆ…c0 4321   class class class wbr 5147  Oncon0 6363  Lim wlim 6364  suc csuc 6365  β€˜cfv 6542  Ο‰com 7857   β‰Ί csdm 8940  cardccrd 9932  β„΅cale 9933  cfccf 9934  Inaccwcwina 10679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-har 9554  df-card 9936  df-aleph 9937  df-cf 9938  df-wina 10681
This theorem is referenced by:  winafp  10694
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