MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tskcard Structured version   Visualization version   GIF version
Theorem tskcard 10812
Description: An even more direct relationship than r1tskina 10813 to get an inaccessible cardinal out of a Tarski class: the size of any nonempty Tarski class is an inaccessible cardinal. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
tskcard ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cardβ€˜π‘‡) ∈ Inacc)
Proof of Theorem tskcard
Dummy variables 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardeq0 10583 . . . 4 (𝑇 ∈ Tarski β†’ ((cardβ€˜π‘‡) = βˆ… ↔ 𝑇 = βˆ…))
21necon3bid 2982 . . 3 (𝑇 ∈ Tarski β†’ ((cardβ€˜π‘‡) β‰  βˆ… ↔ 𝑇 β‰  βˆ…))
32biimpar 476 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cardβ€˜π‘‡) β‰  βˆ…)
4 eqid 2728 . . . . . 6 (𝑧 ∈ (cfβ€˜(β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)})) ↦ (harβ€˜(π‘€β€˜π‘§))) = (𝑧 ∈ (cfβ€˜(β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)})) ↦ (harβ€˜(π‘€β€˜π‘§)))
54pwcfsdom 10614 . . . . 5 (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)}) β‰Ί ((β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)}) ↑m (cfβ€˜(β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)})))
6 vpwex 5381 . . . . . . . . . . . 12 𝒫 π‘₯ ∈ V
76canth2 9161 . . . . . . . . . . 11 𝒫 π‘₯ β‰Ί 𝒫 𝒫 π‘₯
8 simpl 481 . . . . . . . . . . . . 13 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝑇 ∈ Tarski)
9 cardon 9975 . . . . . . . . . . . . . . . . 17 (cardβ€˜π‘‡) ∈ On
109oneli 6488 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ (cardβ€˜π‘‡) β†’ π‘₯ ∈ On)
1110adantl 480 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ π‘₯ ∈ On)
12 cardsdomelir 10004 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ (cardβ€˜π‘‡) β†’ π‘₯ β‰Ί 𝑇)
1312adantl 480 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ π‘₯ β‰Ί 𝑇)
14 tskord 10811 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝑇) β†’ π‘₯ ∈ 𝑇)
158, 11, 13, 14syl3anc 1368 . . . . . . . . . . . . . 14 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ π‘₯ ∈ 𝑇)
16 tskpw 10784 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ 𝑇) β†’ 𝒫 π‘₯ ∈ 𝑇)
17 tskpwss 10783 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ 𝒫 π‘₯ ∈ 𝑇) β†’ 𝒫 𝒫 π‘₯ βŠ† 𝑇)
1816, 17syldan 589 . . . . . . . . . . . . . 14 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ 𝑇) β†’ 𝒫 𝒫 π‘₯ βŠ† 𝑇)
1915, 18syldan 589 . . . . . . . . . . . . 13 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝒫 𝒫 π‘₯ βŠ† 𝑇)
20 ssdomg 9027 . . . . . . . . . . . . 13 (𝑇 ∈ Tarski β†’ (𝒫 𝒫 π‘₯ βŠ† 𝑇 β†’ 𝒫 𝒫 π‘₯ β‰Ό 𝑇))
218, 19, 20sylc 65 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝒫 𝒫 π‘₯ β‰Ό 𝑇)
22 cardidg 10579 . . . . . . . . . . . . . 14 (𝑇 ∈ Tarski β†’ (cardβ€˜π‘‡) β‰ˆ 𝑇)
2322ensymd 9032 . . . . . . . . . . . . 13 (𝑇 ∈ Tarski β†’ 𝑇 β‰ˆ (cardβ€˜π‘‡))
2423adantr 479 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝑇 β‰ˆ (cardβ€˜π‘‡))
25 domentr 9040 . . . . . . . . . . . 12 ((𝒫 𝒫 π‘₯ β‰Ό 𝑇 ∧ 𝑇 β‰ˆ (cardβ€˜π‘‡)) β†’ 𝒫 𝒫 π‘₯ β‰Ό (cardβ€˜π‘‡))
2621, 24, 25syl2anc 582 . . . . . . . . . . 11 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝒫 𝒫 π‘₯ β‰Ό (cardβ€˜π‘‡))
27 sdomdomtr 9141 . . . . . . . . . . 11 ((𝒫 π‘₯ β‰Ί 𝒫 𝒫 π‘₯ ∧ 𝒫 𝒫 π‘₯ β‰Ό (cardβ€˜π‘‡)) β†’ 𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡))
287, 26, 27sylancr 585 . . . . . . . . . 10 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡))
2928ralrimiva 3143 . . . . . . . . 9 (𝑇 ∈ Tarski β†’ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡))
3029adantr 479 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡))
31 inawinalem 10720 . . . . . . . . . 10 ((cardβ€˜π‘‡) ∈ On β†’ (βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡) β†’ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)βˆƒπ‘¦ ∈ (cardβ€˜π‘‡)π‘₯ β‰Ί 𝑦))
329, 31ax-mp 5 . . . . . . . . 9 (βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡) β†’ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)βˆƒπ‘¦ ∈ (cardβ€˜π‘‡)π‘₯ β‰Ί 𝑦)
33 winainflem 10724 . . . . . . . . . 10 (((cardβ€˜π‘‡) β‰  βˆ… ∧ (cardβ€˜π‘‡) ∈ On ∧ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)βˆƒπ‘¦ ∈ (cardβ€˜π‘‡)π‘₯ β‰Ί 𝑦) β†’ Ο‰ βŠ† (cardβ€˜π‘‡))
349, 33mp3an2 1445 . . . . . . . . 9 (((cardβ€˜π‘‡) β‰  βˆ… ∧ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)βˆƒπ‘¦ ∈ (cardβ€˜π‘‡)π‘₯ β‰Ί 𝑦) β†’ Ο‰ βŠ† (cardβ€˜π‘‡))
3532, 34sylan2 591 . . . . . . . 8 (((cardβ€˜π‘‡) β‰  βˆ… ∧ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡)) β†’ Ο‰ βŠ† (cardβ€˜π‘‡))
363, 30, 35syl2anc 582 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ Ο‰ βŠ† (cardβ€˜π‘‡))
37 cardidm 9990 . . . . . . 7 (cardβ€˜(cardβ€˜π‘‡)) = (cardβ€˜π‘‡)
38 cardaleph 10120 . . . . . . 7 ((Ο‰ βŠ† (cardβ€˜π‘‡) ∧ (cardβ€˜(cardβ€˜π‘‡)) = (cardβ€˜π‘‡)) β†’ (cardβ€˜π‘‡) = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)}))
3936, 37, 38sylancl 584 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cardβ€˜π‘‡) = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)}))
4039fveq2d 6906 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cfβ€˜(cardβ€˜π‘‡)) = (cfβ€˜(β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)})))
4139, 40oveq12d 7444 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) = ((β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)}) ↑m (cfβ€˜(β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)}))))
4239, 41breq12d 5165 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ ((cardβ€˜π‘‡) β‰Ί ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) ↔ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)}) β‰Ί ((β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)}) ↑m (cfβ€˜(β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)})))))
435, 42mpbiri 257 . . . 4 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cardβ€˜π‘‡) β‰Ί ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))))
44 simp1 1133 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ 𝑇 ∈ Tarski)
45 simp3 1135 . . . . . . . . . . . . 13 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))))
46 fvex 6915 . . . . . . . . . . . . . . . 16 (cardβ€˜π‘‡) ∈ V
47 fvex 6915 . . . . . . . . . . . . . . . 16 (cfβ€˜(cardβ€˜π‘‡)) ∈ V
4846, 47elmap 8896 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) ↔ π‘₯:(cfβ€˜(cardβ€˜π‘‡))⟢(cardβ€˜π‘‡))
49 fssxp 6756 . . . . . . . . . . . . . . 15 (π‘₯:(cfβ€˜(cardβ€˜π‘‡))⟢(cardβ€˜π‘‡) β†’ π‘₯ βŠ† ((cfβ€˜(cardβ€˜π‘‡)) Γ— (cardβ€˜π‘‡)))
5048, 49sylbi 216 . . . . . . . . . . . . . 14 (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β†’ π‘₯ βŠ† ((cfβ€˜(cardβ€˜π‘‡)) Γ— (cardβ€˜π‘‡)))
5115ex 411 . . . . . . . . . . . . . . . 16 (𝑇 ∈ Tarski β†’ (π‘₯ ∈ (cardβ€˜π‘‡) β†’ π‘₯ ∈ 𝑇))
5251ssrdv 3988 . . . . . . . . . . . . . . 15 (𝑇 ∈ Tarski β†’ (cardβ€˜π‘‡) βŠ† 𝑇)
53 cfle 10285 . . . . . . . . . . . . . . . . 17 (cfβ€˜(cardβ€˜π‘‡)) βŠ† (cardβ€˜π‘‡)
54 sstr 3990 . . . . . . . . . . . . . . . . 17 (((cfβ€˜(cardβ€˜π‘‡)) βŠ† (cardβ€˜π‘‡) ∧ (cardβ€˜π‘‡) βŠ† 𝑇) β†’ (cfβ€˜(cardβ€˜π‘‡)) βŠ† 𝑇)
5553, 54mpan 688 . . . . . . . . . . . . . . . 16 ((cardβ€˜π‘‡) βŠ† 𝑇 β†’ (cfβ€˜(cardβ€˜π‘‡)) βŠ† 𝑇)
56 tskxpss 10803 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) βŠ† 𝑇 ∧ (cardβ€˜π‘‡) βŠ† 𝑇) β†’ ((cfβ€˜(cardβ€˜π‘‡)) Γ— (cardβ€˜π‘‡)) βŠ† 𝑇)
57563exp 1116 . . . . . . . . . . . . . . . . 17 (𝑇 ∈ Tarski β†’ ((cfβ€˜(cardβ€˜π‘‡)) βŠ† 𝑇 β†’ ((cardβ€˜π‘‡) βŠ† 𝑇 β†’ ((cfβ€˜(cardβ€˜π‘‡)) Γ— (cardβ€˜π‘‡)) βŠ† 𝑇)))
5857com23 86 . . . . . . . . . . . . . . . 16 (𝑇 ∈ Tarski β†’ ((cardβ€˜π‘‡) βŠ† 𝑇 β†’ ((cfβ€˜(cardβ€˜π‘‡)) βŠ† 𝑇 β†’ ((cfβ€˜(cardβ€˜π‘‡)) Γ— (cardβ€˜π‘‡)) βŠ† 𝑇)))
5955, 58mpdi 45 . . . . . . . . . . . . . . 15 (𝑇 ∈ Tarski β†’ ((cardβ€˜π‘‡) βŠ† 𝑇 β†’ ((cfβ€˜(cardβ€˜π‘‡)) Γ— (cardβ€˜π‘‡)) βŠ† 𝑇))
6052, 59mpd 15 . . . . . . . . . . . . . 14 (𝑇 ∈ Tarski β†’ ((cfβ€˜(cardβ€˜π‘‡)) Γ— (cardβ€˜π‘‡)) βŠ† 𝑇)
61 sstr2 3989 . . . . . . . . . . . . . 14 (π‘₯ βŠ† ((cfβ€˜(cardβ€˜π‘‡)) Γ— (cardβ€˜π‘‡)) β†’ (((cfβ€˜(cardβ€˜π‘‡)) Γ— (cardβ€˜π‘‡)) βŠ† 𝑇 β†’ π‘₯ βŠ† 𝑇))
6250, 60, 61syl2im 40 . . . . . . . . . . . . 13 (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β†’ (𝑇 ∈ Tarski β†’ π‘₯ βŠ† 𝑇))
6345, 44, 62sylc 65 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ π‘₯ βŠ† 𝑇)
64 simp2 1134 . . . . . . . . . . . . 13 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡))
65 ffn 6727 . . . . . . . . . . . . . . . . 17 (π‘₯:(cfβ€˜(cardβ€˜π‘‡))⟢(cardβ€˜π‘‡) β†’ π‘₯ Fn (cfβ€˜(cardβ€˜π‘‡)))
66 fndmeng 9066 . . . . . . . . . . . . . . . . 17 ((π‘₯ Fn (cfβ€˜(cardβ€˜π‘‡)) ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ V) β†’ (cfβ€˜(cardβ€˜π‘‡)) β‰ˆ π‘₯)
6765, 47, 66sylancl 584 . . . . . . . . . . . . . . . 16 (π‘₯:(cfβ€˜(cardβ€˜π‘‡))⟢(cardβ€˜π‘‡) β†’ (cfβ€˜(cardβ€˜π‘‡)) β‰ˆ π‘₯)
6848, 67sylbi 216 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β†’ (cfβ€˜(cardβ€˜π‘‡)) β‰ˆ π‘₯)
6968ensymd 9032 . . . . . . . . . . . . . 14 (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β†’ π‘₯ β‰ˆ (cfβ€˜(cardβ€˜π‘‡)))
70 cardsdomelir 10004 . . . . . . . . . . . . . 14 ((cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) β†’ (cfβ€˜(cardβ€˜π‘‡)) β‰Ί 𝑇)
71 ensdomtr 9144 . . . . . . . . . . . . . 14 ((π‘₯ β‰ˆ (cfβ€˜(cardβ€˜π‘‡)) ∧ (cfβ€˜(cardβ€˜π‘‡)) β‰Ί 𝑇) β†’ π‘₯ β‰Ί 𝑇)
7269, 70, 71syl2an 594 . . . . . . . . . . . . 13 ((π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ π‘₯ β‰Ί 𝑇)
7345, 64, 72syl2anc 582 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ π‘₯ β‰Ί 𝑇)
74 tskssel 10788 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ π‘₯ βŠ† 𝑇 ∧ π‘₯ β‰Ί 𝑇) β†’ π‘₯ ∈ 𝑇)
7544, 63, 73, 74syl3anc 1368 . . . . . . . . . . 11 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ π‘₯ ∈ 𝑇)
76753expia 1118 . . . . . . . . . 10 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β†’ π‘₯ ∈ 𝑇))
7776ssrdv 3988 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) βŠ† 𝑇)
78 ssdomg 9027 . . . . . . . . . 10 (𝑇 ∈ Tarski β†’ (((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) βŠ† 𝑇 β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό 𝑇))
7978imp 405 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) βŠ† 𝑇) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό 𝑇)
8077, 79syldan 589 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό 𝑇)
8123adantr 479 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ 𝑇 β‰ˆ (cardβ€˜π‘‡))
82 domentr 9040 . . . . . . . 8 ((((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό 𝑇 ∧ 𝑇 β‰ˆ (cardβ€˜π‘‡)) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό (cardβ€˜π‘‡))
8380, 81, 82syl2anc 582 . . . . . . 7 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό (cardβ€˜π‘‡))
84 domnsym 9130 . . . . . . 7 (((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό (cardβ€˜π‘‡) β†’ Β¬ (cardβ€˜π‘‡) β‰Ί ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))))
8583, 84syl 17 . . . . . 6 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ Β¬ (cardβ€˜π‘‡) β‰Ί ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))))
8685ex 411 . . . . 5 (𝑇 ∈ Tarski β†’ ((cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) β†’ Β¬ (cardβ€˜π‘‡) β‰Ί ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))))
8786adantr 479 . . . 4 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ ((cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) β†’ Β¬ (cardβ€˜π‘‡) β‰Ί ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))))
8843, 87mt2d 136 . . 3 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ Β¬ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡))
89 cfon 10286 . . . . . 6 (cfβ€˜(cardβ€˜π‘‡)) ∈ On
9089, 9onsseli 6495 . . . . 5 ((cfβ€˜(cardβ€˜π‘‡)) βŠ† (cardβ€˜π‘‡) ↔ ((cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∨ (cfβ€˜(cardβ€˜π‘‡)) = (cardβ€˜π‘‡)))
9153, 90mpbi 229 . . . 4 ((cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∨ (cfβ€˜(cardβ€˜π‘‡)) = (cardβ€˜π‘‡))
9291ori 859 . . 3 (Β¬ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) β†’ (cfβ€˜(cardβ€˜π‘‡)) = (cardβ€˜π‘‡))
9388, 92syl 17 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cfβ€˜(cardβ€˜π‘‡)) = (cardβ€˜π‘‡))
94 elina 10718 . 2 ((cardβ€˜π‘‡) ∈ Inacc ↔ ((cardβ€˜π‘‡) β‰  βˆ… ∧ (cfβ€˜(cardβ€˜π‘‡)) = (cardβ€˜π‘‡) ∧ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡)))
953, 93, 30, 94syl3anbrc 1340 1 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cardβ€˜π‘‡) ∈ Inacc)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆ€wral 3058  βˆƒwrex 3067  {crab 3430  Vcvv 3473   βŠ† wss 3949  βˆ…c0 4326  π’« cpw 4606  βˆ© cint 4953   class class class wbr 5152   ↦ cmpt 5235   Γ— cxp 5680  Oncon0 6374   Fn wfn 6548  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  Ο‰com 7876   ↑m cmap 8851   β‰ˆ cen 8967   β‰Ό cdom 8968   β‰Ί csdm 8969  harchar 9587  cardccrd 9966  β„΅cale 9967  cfccf 9968  Inacccina 10714  Tarskictsk 10779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-ac2 10494
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  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 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-smo 8373  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-er 8731  df-map 8853  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-oi 9541  df-har 9588  df-r1 9795  df-card 9970  df-aleph 9971  df-cf 9972  df-acn 9973  df-ac 10147  df-ina 10716  df-tsk 10780
This theorem is referenced by:  r1tskina  10813  tskuni  10814  inaprc  10867
  Copyright terms: Public domain W3C validator