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Theorem tskcard 10776
Description: An even more direct relationship than r1tskina 10777 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 10547 . . . 4 (𝑇 ∈ Tarski β†’ ((cardβ€˜π‘‡) = βˆ… ↔ 𝑇 = βˆ…))
21necon3bid 2986 . . 3 (𝑇 ∈ Tarski β†’ ((cardβ€˜π‘‡) β‰  βˆ… ↔ 𝑇 β‰  βˆ…))
32biimpar 479 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cardβ€˜π‘‡) β‰  βˆ…)
4 eqid 2733 . . . . . 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 10578 . . . . 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 5376 . . . . . . . . . . . 12 𝒫 π‘₯ ∈ V
76canth2 9130 . . . . . . . . . . 11 𝒫 π‘₯ β‰Ί 𝒫 𝒫 π‘₯
8 simpl 484 . . . . . . . . . . . . 13 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝑇 ∈ Tarski)
9 cardon 9939 . . . . . . . . . . . . . . . . 17 (cardβ€˜π‘‡) ∈ On
109oneli 6479 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ (cardβ€˜π‘‡) β†’ π‘₯ ∈ On)
1110adantl 483 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ π‘₯ ∈ On)
12 cardsdomelir 9968 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ (cardβ€˜π‘‡) β†’ π‘₯ β‰Ί 𝑇)
1312adantl 483 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ π‘₯ β‰Ί 𝑇)
14 tskord 10775 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝑇) β†’ π‘₯ ∈ 𝑇)
158, 11, 13, 14syl3anc 1372 . . . . . . . . . . . . . 14 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ π‘₯ ∈ 𝑇)
16 tskpw 10748 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ 𝑇) β†’ 𝒫 π‘₯ ∈ 𝑇)
17 tskpwss 10747 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ 𝒫 π‘₯ ∈ 𝑇) β†’ 𝒫 𝒫 π‘₯ βŠ† 𝑇)
1816, 17syldan 592 . . . . . . . . . . . . . 14 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ 𝑇) β†’ 𝒫 𝒫 π‘₯ βŠ† 𝑇)
1915, 18syldan 592 . . . . . . . . . . . . 13 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝒫 𝒫 π‘₯ βŠ† 𝑇)
20 ssdomg 8996 . . . . . . . . . . . . 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 10543 . . . . . . . . . . . . . 14 (𝑇 ∈ Tarski β†’ (cardβ€˜π‘‡) β‰ˆ 𝑇)
2322ensymd 9001 . . . . . . . . . . . . 13 (𝑇 ∈ Tarski β†’ 𝑇 β‰ˆ (cardβ€˜π‘‡))
2423adantr 482 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝑇 β‰ˆ (cardβ€˜π‘‡))
25 domentr 9009 . . . . . . . . . . . 12 ((𝒫 𝒫 π‘₯ β‰Ό 𝑇 ∧ 𝑇 β‰ˆ (cardβ€˜π‘‡)) β†’ 𝒫 𝒫 π‘₯ β‰Ό (cardβ€˜π‘‡))
2621, 24, 25syl2anc 585 . . . . . . . . . . 11 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝒫 𝒫 π‘₯ β‰Ό (cardβ€˜π‘‡))
27 sdomdomtr 9110 . . . . . . . . . . 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 588 . . . . . . . . . 10 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡))
2928ralrimiva 3147 . . . . . . . . 9 (𝑇 ∈ Tarski β†’ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡))
3029adantr 482 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡))
31 inawinalem 10684 . . . . . . . . . 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 10688 . . . . . . . . . 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 1450 . . . . . . . . 9 (((cardβ€˜π‘‡) β‰  βˆ… ∧ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)βˆƒπ‘¦ ∈ (cardβ€˜π‘‡)π‘₯ β‰Ί 𝑦) β†’ Ο‰ βŠ† (cardβ€˜π‘‡))
3532, 34sylan2 594 . . . . . . . 8 (((cardβ€˜π‘‡) β‰  βˆ… ∧ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡)) β†’ Ο‰ βŠ† (cardβ€˜π‘‡))
363, 30, 35syl2anc 585 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ Ο‰ βŠ† (cardβ€˜π‘‡))
37 cardidm 9954 . . . . . . 7 (cardβ€˜(cardβ€˜π‘‡)) = (cardβ€˜π‘‡)
38 cardaleph 10084 . . . . . . 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 587 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cardβ€˜π‘‡) = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)}))
4039fveq2d 6896 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cfβ€˜(cardβ€˜π‘‡)) = (cfβ€˜(β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)})))
4139, 40oveq12d 7427 . . . . . 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 5162 . . . . 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 258 . . . 4 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cardβ€˜π‘‡) β‰Ί ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))))
44 simp1 1137 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ 𝑇 ∈ Tarski)
45 simp3 1139 . . . . . . . . . . . . 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 6905 . . . . . . . . . . . . . . . 16 (cardβ€˜π‘‡) ∈ V
47 fvex 6905 . . . . . . . . . . . . . . . 16 (cfβ€˜(cardβ€˜π‘‡)) ∈ V
4846, 47elmap 8865 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) ↔ π‘₯:(cfβ€˜(cardβ€˜π‘‡))⟢(cardβ€˜π‘‡))
49 fssxp 6746 . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . 16 (𝑇 ∈ Tarski β†’ (π‘₯ ∈ (cardβ€˜π‘‡) β†’ π‘₯ ∈ 𝑇))
5251ssrdv 3989 . . . . . . . . . . . . . . 15 (𝑇 ∈ Tarski β†’ (cardβ€˜π‘‡) βŠ† 𝑇)
53 cfle 10249 . . . . . . . . . . . . . . . . 17 (cfβ€˜(cardβ€˜π‘‡)) βŠ† (cardβ€˜π‘‡)
54 sstr 3991 . . . . . . . . . . . . . . . . 17 (((cfβ€˜(cardβ€˜π‘‡)) βŠ† (cardβ€˜π‘‡) ∧ (cardβ€˜π‘‡) βŠ† 𝑇) β†’ (cfβ€˜(cardβ€˜π‘‡)) βŠ† 𝑇)
5553, 54mpan 689 . . . . . . . . . . . . . . . 16 ((cardβ€˜π‘‡) βŠ† 𝑇 β†’ (cfβ€˜(cardβ€˜π‘‡)) βŠ† 𝑇)
56 tskxpss 10767 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) βŠ† 𝑇 ∧ (cardβ€˜π‘‡) βŠ† 𝑇) β†’ ((cfβ€˜(cardβ€˜π‘‡)) Γ— (cardβ€˜π‘‡)) βŠ† 𝑇)
57563exp 1120 . . . . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . 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 1138 . . . . . . . . . . . . 13 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡))
65 ffn 6718 . . . . . . . . . . . . . . . . 17 (π‘₯:(cfβ€˜(cardβ€˜π‘‡))⟢(cardβ€˜π‘‡) β†’ π‘₯ Fn (cfβ€˜(cardβ€˜π‘‡)))
66 fndmeng 9035 . . . . . . . . . . . . . . . . 17 ((π‘₯ Fn (cfβ€˜(cardβ€˜π‘‡)) ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ V) β†’ (cfβ€˜(cardβ€˜π‘‡)) β‰ˆ π‘₯)
6765, 47, 66sylancl 587 . . . . . . . . . . . . . . . 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 9001 . . . . . . . . . . . . . 14 (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β†’ π‘₯ β‰ˆ (cfβ€˜(cardβ€˜π‘‡)))
70 cardsdomelir 9968 . . . . . . . . . . . . . 14 ((cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) β†’ (cfβ€˜(cardβ€˜π‘‡)) β‰Ί 𝑇)
71 ensdomtr 9113 . . . . . . . . . . . . . 14 ((π‘₯ β‰ˆ (cfβ€˜(cardβ€˜π‘‡)) ∧ (cfβ€˜(cardβ€˜π‘‡)) β‰Ί 𝑇) β†’ π‘₯ β‰Ί 𝑇)
7269, 70, 71syl2an 597 . . . . . . . . . . . . 13 ((π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ π‘₯ β‰Ί 𝑇)
7345, 64, 72syl2anc 585 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ π‘₯ β‰Ί 𝑇)
74 tskssel 10752 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ π‘₯ βŠ† 𝑇 ∧ π‘₯ β‰Ί 𝑇) β†’ π‘₯ ∈ 𝑇)
7544, 63, 73, 74syl3anc 1372 . . . . . . . . . . 11 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ π‘₯ ∈ 𝑇)
76753expia 1122 . . . . . . . . . 10 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β†’ π‘₯ ∈ 𝑇))
7776ssrdv 3989 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) βŠ† 𝑇)
78 ssdomg 8996 . . . . . . . . . 10 (𝑇 ∈ Tarski β†’ (((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) βŠ† 𝑇 β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό 𝑇))
7978imp 408 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) βŠ† 𝑇) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό 𝑇)
8077, 79syldan 592 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό 𝑇)
8123adantr 482 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ 𝑇 β‰ˆ (cardβ€˜π‘‡))
82 domentr 9009 . . . . . . . 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 585 . . . . . . 7 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό (cardβ€˜π‘‡))
84 domnsym 9099 . . . . . . 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 414 . . . . 5 (𝑇 ∈ Tarski β†’ ((cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) β†’ Β¬ (cardβ€˜π‘‡) β‰Ί ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))))
8786adantr 482 . . . 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 10250 . . . . . 6 (cfβ€˜(cardβ€˜π‘‡)) ∈ On
9089, 9onsseli 6486 . . . . 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 860 . . 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 10682 . 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 1344 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆ© cint 4951   class class class wbr 5149   ↦ cmpt 5232   Γ— cxp 5675  Oncon0 6365   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Ο‰com 7855   ↑m cmap 8820   β‰ˆ cen 8936   β‰Ό cdom 8937   β‰Ί csdm 8938  harchar 9551  cardccrd 9930  β„΅cale 9931  cfccf 9932  Inacccina 10678  Tarskictsk 10743
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-ac2 10458
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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-iin 5001  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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-smo 8346  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-har 9552  df-r1 9759  df-card 9934  df-aleph 9935  df-cf 9936  df-acn 9937  df-ac 10111  df-ina 10680  df-tsk 10744
This theorem is referenced by:  r1tskina  10777  tskuni  10778  inaprc  10831
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