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Theorem tskcard 10775
Description: An even more direct relationship than r1tskina 10776 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 10546 . . . 4 (𝑇 ∈ Tarski β†’ ((cardβ€˜π‘‡) = βˆ… ↔ 𝑇 = βˆ…))
21necon3bid 2979 . . 3 (𝑇 ∈ Tarski β†’ ((cardβ€˜π‘‡) β‰  βˆ… ↔ 𝑇 β‰  βˆ…))
32biimpar 477 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cardβ€˜π‘‡) β‰  βˆ…)
4 eqid 2726 . . . . . 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 10577 . . . . 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 5368 . . . . . . . . . . . 12 𝒫 π‘₯ ∈ V
76canth2 9129 . . . . . . . . . . 11 𝒫 π‘₯ β‰Ί 𝒫 𝒫 π‘₯
8 simpl 482 . . . . . . . . . . . . 13 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝑇 ∈ Tarski)
9 cardon 9938 . . . . . . . . . . . . . . . . 17 (cardβ€˜π‘‡) ∈ On
109oneli 6471 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ (cardβ€˜π‘‡) β†’ π‘₯ ∈ On)
1110adantl 481 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ π‘₯ ∈ On)
12 cardsdomelir 9967 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ (cardβ€˜π‘‡) β†’ π‘₯ β‰Ί 𝑇)
1312adantl 481 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ π‘₯ β‰Ί 𝑇)
14 tskord 10774 . . . . . . . . . . . . . . 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 10747 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ 𝑇) β†’ 𝒫 π‘₯ ∈ 𝑇)
17 tskpwss 10746 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Tarski ∧ 𝒫 π‘₯ ∈ 𝑇) β†’ 𝒫 𝒫 π‘₯ βŠ† 𝑇)
1816, 17syldan 590 . . . . . . . . . . . . . 14 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ 𝑇) β†’ 𝒫 𝒫 π‘₯ βŠ† 𝑇)
1915, 18syldan 590 . . . . . . . . . . . . 13 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝒫 𝒫 π‘₯ βŠ† 𝑇)
20 ssdomg 8995 . . . . . . . . . . . . 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 10542 . . . . . . . . . . . . . 14 (𝑇 ∈ Tarski β†’ (cardβ€˜π‘‡) β‰ˆ 𝑇)
2322ensymd 9000 . . . . . . . . . . . . 13 (𝑇 ∈ Tarski β†’ 𝑇 β‰ˆ (cardβ€˜π‘‡))
2423adantr 480 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝑇 β‰ˆ (cardβ€˜π‘‡))
25 domentr 9008 . . . . . . . . . . . 12 ((𝒫 𝒫 π‘₯ β‰Ό 𝑇 ∧ 𝑇 β‰ˆ (cardβ€˜π‘‡)) β†’ 𝒫 𝒫 π‘₯ β‰Ό (cardβ€˜π‘‡))
2621, 24, 25syl2anc 583 . . . . . . . . . . 11 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝒫 𝒫 π‘₯ β‰Ό (cardβ€˜π‘‡))
27 sdomdomtr 9109 . . . . . . . . . . 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 586 . . . . . . . . . 10 ((𝑇 ∈ Tarski ∧ π‘₯ ∈ (cardβ€˜π‘‡)) β†’ 𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡))
2928ralrimiva 3140 . . . . . . . . 9 (𝑇 ∈ Tarski β†’ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡))
3029adantr 480 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡))
31 inawinalem 10683 . . . . . . . . . 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 10687 . . . . . . . . . 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 592 . . . . . . . 8 (((cardβ€˜π‘‡) β‰  βˆ… ∧ βˆ€π‘₯ ∈ (cardβ€˜π‘‡)𝒫 π‘₯ β‰Ί (cardβ€˜π‘‡)) β†’ Ο‰ βŠ† (cardβ€˜π‘‡))
363, 30, 35syl2anc 583 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ Ο‰ βŠ† (cardβ€˜π‘‡))
37 cardidm 9953 . . . . . . 7 (cardβ€˜(cardβ€˜π‘‡)) = (cardβ€˜π‘‡)
38 cardaleph 10083 . . . . . . 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 585 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cardβ€˜π‘‡) = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)}))
4039fveq2d 6888 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑇 β‰  βˆ…) β†’ (cfβ€˜(cardβ€˜π‘‡)) = (cfβ€˜(β„΅β€˜βˆ© {π‘₯ ∈ On ∣ (cardβ€˜π‘‡) βŠ† (β„΅β€˜π‘₯)})))
4139, 40oveq12d 7422 . . . . . 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 5154 . . . . 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 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 6897 . . . . . . . . . . . . . . . 16 (cardβ€˜π‘‡) ∈ V
47 fvex 6897 . . . . . . . . . . . . . . . 16 (cfβ€˜(cardβ€˜π‘‡)) ∈ V
4846, 47elmap 8864 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) ↔ π‘₯:(cfβ€˜(cardβ€˜π‘‡))⟢(cardβ€˜π‘‡))
49 fssxp 6738 . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . 16 (𝑇 ∈ Tarski β†’ (π‘₯ ∈ (cardβ€˜π‘‡) β†’ π‘₯ ∈ 𝑇))
5251ssrdv 3983 . . . . . . . . . . . . . . 15 (𝑇 ∈ Tarski β†’ (cardβ€˜π‘‡) βŠ† 𝑇)
53 cfle 10248 . . . . . . . . . . . . . . . . 17 (cfβ€˜(cardβ€˜π‘‡)) βŠ† (cardβ€˜π‘‡)
54 sstr 3985 . . . . . . . . . . . . . . . . 17 (((cfβ€˜(cardβ€˜π‘‡)) βŠ† (cardβ€˜π‘‡) ∧ (cardβ€˜π‘‡) βŠ† 𝑇) β†’ (cfβ€˜(cardβ€˜π‘‡)) βŠ† 𝑇)
5553, 54mpan 687 . . . . . . . . . . . . . . . 16 ((cardβ€˜π‘‡) βŠ† 𝑇 β†’ (cfβ€˜(cardβ€˜π‘‡)) βŠ† 𝑇)
56 tskxpss 10766 . . . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . 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 6710 . . . . . . . . . . . . . . . . 17 (π‘₯:(cfβ€˜(cardβ€˜π‘‡))⟢(cardβ€˜π‘‡) β†’ π‘₯ Fn (cfβ€˜(cardβ€˜π‘‡)))
66 fndmeng 9034 . . . . . . . . . . . . . . . . 17 ((π‘₯ Fn (cfβ€˜(cardβ€˜π‘‡)) ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ V) β†’ (cfβ€˜(cardβ€˜π‘‡)) β‰ˆ π‘₯)
6765, 47, 66sylancl 585 . . . . . . . . . . . . . . . 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 9000 . . . . . . . . . . . . . 14 (π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β†’ π‘₯ β‰ˆ (cfβ€˜(cardβ€˜π‘‡)))
70 cardsdomelir 9967 . . . . . . . . . . . . . 14 ((cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) β†’ (cfβ€˜(cardβ€˜π‘‡)) β‰Ί 𝑇)
71 ensdomtr 9112 . . . . . . . . . . . . . 14 ((π‘₯ β‰ˆ (cfβ€˜(cardβ€˜π‘‡)) ∧ (cfβ€˜(cardβ€˜π‘‡)) β‰Ί 𝑇) β†’ π‘₯ β‰Ί 𝑇)
7269, 70, 71syl2an 595 . . . . . . . . . . . . 13 ((π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ π‘₯ β‰Ί 𝑇)
7345, 64, 72syl2anc 583 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) ∧ π‘₯ ∈ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))) β†’ π‘₯ β‰Ί 𝑇)
74 tskssel 10751 . . . . . . . . . . . 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 3983 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) βŠ† 𝑇)
78 ssdomg 8995 . . . . . . . . . 10 (𝑇 ∈ Tarski β†’ (((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) βŠ† 𝑇 β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό 𝑇))
7978imp 406 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) βŠ† 𝑇) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό 𝑇)
8077, 79syldan 590 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό 𝑇)
8123adantr 480 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ 𝑇 β‰ˆ (cardβ€˜π‘‡))
82 domentr 9008 . . . . . . . 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 583 . . . . . . 7 ((𝑇 ∈ Tarski ∧ (cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡)) β†’ ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡))) β‰Ό (cardβ€˜π‘‡))
84 domnsym 9098 . . . . . . 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 412 . . . . 5 (𝑇 ∈ Tarski β†’ ((cfβ€˜(cardβ€˜π‘‡)) ∈ (cardβ€˜π‘‡) β†’ Β¬ (cardβ€˜π‘‡) β‰Ί ((cardβ€˜π‘‡) ↑m (cfβ€˜(cardβ€˜π‘‡)))))
8786adantr 480 . . . 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 10249 . . . . . 6 (cfβ€˜(cardβ€˜π‘‡)) ∈ On
9089, 9onsseli 6478 . . . . 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 858 . . 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 10681 . 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 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  {crab 3426  Vcvv 3468   βŠ† wss 3943  βˆ…c0 4317  π’« cpw 4597  βˆ© cint 4943   class class class wbr 5141   ↦ cmpt 5224   Γ— cxp 5667  Oncon0 6357   Fn wfn 6531  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  Ο‰com 7851   ↑m cmap 8819   β‰ˆ cen 8935   β‰Ό cdom 8936   β‰Ί csdm 8937  harchar 9550  cardccrd 9929  β„΅cale 9930  cfccf 9931  Inacccina 10677  Tarskictsk 10742
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-inf2 9635  ax-ac2 10457
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-smo 8344  df-recs 8369  df-rdg 8408  df-1o 8464  df-2o 8465  df-er 8702  df-map 8821  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-oi 9504  df-har 9551  df-r1 9758  df-card 9933  df-aleph 9934  df-cf 9935  df-acn 9936  df-ac 10110  df-ina 10679  df-tsk 10743
This theorem is referenced by:  r1tskina  10776  tskuni  10777  inaprc  10830
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