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Theorem cfsuc 10252
Description: Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
Assertion
Ref Expression
cfsuc (𝐴 ∈ On β†’ (cfβ€˜suc 𝐴) = 1o)

Proof of Theorem cfsuc
Dummy variables π‘₯ 𝑦 𝑧 𝑀 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onsucb 7805 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
2 cfval 10242 . . 3 (suc 𝐴 ∈ On β†’ (cfβ€˜suc 𝐴) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
31, 2sylbi 216 . 2 (𝐴 ∈ On β†’ (cfβ€˜suc 𝐴) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
4 cardsn 9964 . . . . . 6 (𝐴 ∈ On β†’ (cardβ€˜{𝐴}) = 1o)
54eqcomd 2739 . . . . 5 (𝐴 ∈ On β†’ 1o = (cardβ€˜{𝐴}))
6 snidg 4663 . . . . . . . 8 (𝐴 ∈ On β†’ 𝐴 ∈ {𝐴})
7 elsuci 6432 . . . . . . . . 9 (𝑧 ∈ suc 𝐴 β†’ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴))
8 onelss 6407 . . . . . . . . . 10 (𝐴 ∈ On β†’ (𝑧 ∈ 𝐴 β†’ 𝑧 βŠ† 𝐴))
9 eqimss 4041 . . . . . . . . . . 11 (𝑧 = 𝐴 β†’ 𝑧 βŠ† 𝐴)
109a1i 11 . . . . . . . . . 10 (𝐴 ∈ On β†’ (𝑧 = 𝐴 β†’ 𝑧 βŠ† 𝐴))
118, 10jaod 858 . . . . . . . . 9 (𝐴 ∈ On β†’ ((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴) β†’ 𝑧 βŠ† 𝐴))
127, 11syl5 34 . . . . . . . 8 (𝐴 ∈ On β†’ (𝑧 ∈ suc 𝐴 β†’ 𝑧 βŠ† 𝐴))
13 sseq2 4009 . . . . . . . . 9 (𝑀 = 𝐴 β†’ (𝑧 βŠ† 𝑀 ↔ 𝑧 βŠ† 𝐴))
1413rspcev 3613 . . . . . . . 8 ((𝐴 ∈ {𝐴} ∧ 𝑧 βŠ† 𝐴) β†’ βˆƒπ‘€ ∈ {𝐴}𝑧 βŠ† 𝑀)
156, 12, 14syl6an 683 . . . . . . 7 (𝐴 ∈ On β†’ (𝑧 ∈ suc 𝐴 β†’ βˆƒπ‘€ ∈ {𝐴}𝑧 βŠ† 𝑀))
1615ralrimiv 3146 . . . . . 6 (𝐴 ∈ On β†’ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ {𝐴}𝑧 βŠ† 𝑀)
17 ssun2 4174 . . . . . . 7 {𝐴} βŠ† (𝐴 βˆͺ {𝐴})
18 df-suc 6371 . . . . . . 7 suc 𝐴 = (𝐴 βˆͺ {𝐴})
1917, 18sseqtrri 4020 . . . . . 6 {𝐴} βŠ† suc 𝐴
2016, 19jctil 521 . . . . 5 (𝐴 ∈ On β†’ ({𝐴} βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ {𝐴}𝑧 βŠ† 𝑀))
21 snex 5432 . . . . . 6 {𝐴} ∈ V
22 fveq2 6892 . . . . . . . 8 (𝑦 = {𝐴} β†’ (cardβ€˜π‘¦) = (cardβ€˜{𝐴}))
2322eqeq2d 2744 . . . . . . 7 (𝑦 = {𝐴} β†’ (1o = (cardβ€˜π‘¦) ↔ 1o = (cardβ€˜{𝐴})))
24 sseq1 4008 . . . . . . . 8 (𝑦 = {𝐴} β†’ (𝑦 βŠ† suc 𝐴 ↔ {𝐴} βŠ† suc 𝐴))
25 rexeq 3322 . . . . . . . . 9 (𝑦 = {𝐴} β†’ (βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆƒπ‘€ ∈ {𝐴}𝑧 βŠ† 𝑀))
2625ralbidv 3178 . . . . . . . 8 (𝑦 = {𝐴} β†’ (βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ {𝐴}𝑧 βŠ† 𝑀))
2724, 26anbi12d 632 . . . . . . 7 (𝑦 = {𝐴} β†’ ((𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀) ↔ ({𝐴} βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ {𝐴}𝑧 βŠ† 𝑀)))
2823, 27anbi12d 632 . . . . . 6 (𝑦 = {𝐴} β†’ ((1o = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ (1o = (cardβ€˜{𝐴}) ∧ ({𝐴} βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ {𝐴}𝑧 βŠ† 𝑀))))
2921, 28spcev 3597 . . . . 5 ((1o = (cardβ€˜{𝐴}) ∧ ({𝐴} βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ {𝐴}𝑧 βŠ† 𝑀)) β†’ βˆƒπ‘¦(1o = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
305, 20, 29syl2anc 585 . . . 4 (𝐴 ∈ On β†’ βˆƒπ‘¦(1o = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
31 1oex 8476 . . . . 5 1o ∈ V
32 eqeq1 2737 . . . . . . 7 (π‘₯ = 1o β†’ (π‘₯ = (cardβ€˜π‘¦) ↔ 1o = (cardβ€˜π‘¦)))
3332anbi1d 631 . . . . . 6 (π‘₯ = 1o β†’ ((π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ (1o = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
3433exbidv 1925 . . . . 5 (π‘₯ = 1o β†’ (βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ βˆƒπ‘¦(1o = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
3531, 34elab 3669 . . . 4 (1o ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ↔ βˆƒπ‘¦(1o = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
3630, 35sylibr 233 . . 3 (𝐴 ∈ On β†’ 1o ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
37 el1o 8495 . . . . 5 (𝑣 ∈ 1o ↔ 𝑣 = βˆ…)
38 eqcom 2740 . . . . . . . . . . . . . . 15 (βˆ… = (cardβ€˜π‘¦) ↔ (cardβ€˜π‘¦) = βˆ…)
39 vex 3479 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
40 onssnum 10035 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ V ∧ 𝑦 βŠ† On) β†’ 𝑦 ∈ dom card)
4139, 40mpan 689 . . . . . . . . . . . . . . . 16 (𝑦 βŠ† On β†’ 𝑦 ∈ dom card)
42 cardnueq0 9959 . . . . . . . . . . . . . . . 16 (𝑦 ∈ dom card β†’ ((cardβ€˜π‘¦) = βˆ… ↔ 𝑦 = βˆ…))
4341, 42syl 17 . . . . . . . . . . . . . . 15 (𝑦 βŠ† On β†’ ((cardβ€˜π‘¦) = βˆ… ↔ 𝑦 = βˆ…))
4438, 43bitrid 283 . . . . . . . . . . . . . 14 (𝑦 βŠ† On β†’ (βˆ… = (cardβ€˜π‘¦) ↔ 𝑦 = βˆ…))
4544biimpa 478 . . . . . . . . . . . . 13 ((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) β†’ 𝑦 = βˆ…)
46 rex0 4358 . . . . . . . . . . . . . . . . 17 Β¬ βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀
4746a1i 11 . . . . . . . . . . . . . . . 16 (𝑧 ∈ suc 𝐴 β†’ Β¬ βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀)
4847nrex 3075 . . . . . . . . . . . . . . 15 Β¬ βˆƒπ‘§ ∈ suc π΄βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀
49 nsuceq0 6448 . . . . . . . . . . . . . . . 16 suc 𝐴 β‰  βˆ…
50 r19.2z 4495 . . . . . . . . . . . . . . . 16 ((suc 𝐴 β‰  βˆ… ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀) β†’ βˆƒπ‘§ ∈ suc π΄βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀)
5149, 50mpan 689 . . . . . . . . . . . . . . 15 (βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀 β†’ βˆƒπ‘§ ∈ suc π΄βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀)
5248, 51mto 196 . . . . . . . . . . . . . 14 Β¬ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀
53 rexeq 3322 . . . . . . . . . . . . . . 15 (𝑦 = βˆ… β†’ (βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀))
5453ralbidv 3178 . . . . . . . . . . . . . 14 (𝑦 = βˆ… β†’ (βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀))
5552, 54mtbiri 327 . . . . . . . . . . . . 13 (𝑦 = βˆ… β†’ Β¬ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)
5645, 55syl 17 . . . . . . . . . . . 12 ((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) β†’ Β¬ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)
5756intnand 490 . . . . . . . . . . 11 ((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) β†’ Β¬ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
58 imnan 401 . . . . . . . . . . 11 (((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) β†’ Β¬ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ Β¬ ((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
5957, 58mpbi 229 . . . . . . . . . 10 Β¬ ((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
60 onsuc 7799 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On β†’ suc 𝐴 ∈ On)
61 onss 7772 . . . . . . . . . . . . . . . . 17 (suc 𝐴 ∈ On β†’ suc 𝐴 βŠ† On)
62 sstr 3991 . . . . . . . . . . . . . . . . 17 ((𝑦 βŠ† suc 𝐴 ∧ suc 𝐴 βŠ† On) β†’ 𝑦 βŠ† On)
6361, 62sylan2 594 . . . . . . . . . . . . . . . 16 ((𝑦 βŠ† suc 𝐴 ∧ suc 𝐴 ∈ On) β†’ 𝑦 βŠ† On)
6460, 63sylan2 594 . . . . . . . . . . . . . . 15 ((𝑦 βŠ† suc 𝐴 ∧ 𝐴 ∈ On) β†’ 𝑦 βŠ† On)
6564ancoms 460 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 βŠ† suc 𝐴) β†’ 𝑦 βŠ† On)
6665adantrr 716 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ 𝑦 βŠ† On)
67663adant2 1132 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ 𝑦 βŠ† On)
68 simp2 1138 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ βˆ… = (cardβ€˜π‘¦))
69 simp3 1139 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
7067, 68, 69jca31 516 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ ((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
71703expib 1123 . . . . . . . . . 10 (𝐴 ∈ On β†’ ((βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ ((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
7259, 71mtoi 198 . . . . . . . . 9 (𝐴 ∈ On β†’ Β¬ (βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
7372nexdv 1940 . . . . . . . 8 (𝐴 ∈ On β†’ Β¬ βˆƒπ‘¦(βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
74 0ex 5308 . . . . . . . . 9 βˆ… ∈ V
75 eqeq1 2737 . . . . . . . . . . 11 (π‘₯ = βˆ… β†’ (π‘₯ = (cardβ€˜π‘¦) ↔ βˆ… = (cardβ€˜π‘¦)))
7675anbi1d 631 . . . . . . . . . 10 (π‘₯ = βˆ… β†’ ((π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ (βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
7776exbidv 1925 . . . . . . . . 9 (π‘₯ = βˆ… β†’ (βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ βˆƒπ‘¦(βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
7874, 77elab 3669 . . . . . . . 8 (βˆ… ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ↔ βˆƒπ‘¦(βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
7973, 78sylnibr 329 . . . . . . 7 (𝐴 ∈ On β†’ Β¬ βˆ… ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
8079adantr 482 . . . . . 6 ((𝐴 ∈ On ∧ 𝑣 = βˆ…) β†’ Β¬ βˆ… ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
81 eleq1 2822 . . . . . . 7 (𝑣 = βˆ… β†’ (𝑣 ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ↔ βˆ… ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))}))
8281adantl 483 . . . . . 6 ((𝐴 ∈ On ∧ 𝑣 = βˆ…) β†’ (𝑣 ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ↔ βˆ… ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))}))
8380, 82mtbird 325 . . . . 5 ((𝐴 ∈ On ∧ 𝑣 = βˆ…) β†’ Β¬ 𝑣 ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
8437, 83sylan2b 595 . . . 4 ((𝐴 ∈ On ∧ 𝑣 ∈ 1o) β†’ Β¬ 𝑣 ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
8584ralrimiva 3147 . . 3 (𝐴 ∈ On β†’ βˆ€π‘£ ∈ 1o Β¬ 𝑣 ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
86 cardon 9939 . . . . . . . 8 (cardβ€˜π‘¦) ∈ On
87 eleq1 2822 . . . . . . . 8 (π‘₯ = (cardβ€˜π‘¦) β†’ (π‘₯ ∈ On ↔ (cardβ€˜π‘¦) ∈ On))
8886, 87mpbiri 258 . . . . . . 7 (π‘₯ = (cardβ€˜π‘¦) β†’ π‘₯ ∈ On)
8988adantr 482 . . . . . 6 ((π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ π‘₯ ∈ On)
9089exlimiv 1934 . . . . 5 (βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ π‘₯ ∈ On)
9190abssi 4068 . . . 4 {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} βŠ† On
92 oneqmini 6417 . . . 4 ({π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} βŠ† On β†’ ((1o ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ∧ βˆ€π‘£ ∈ 1o Β¬ 𝑣 ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))}) β†’ 1o = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))}))
9391, 92ax-mp 5 . . 3 ((1o ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ∧ βˆ€π‘£ ∈ 1o Β¬ 𝑣 ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))}) β†’ 1o = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
9436, 85, 93syl2anc 585 . 2 (𝐴 ∈ On β†’ 1o = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† suc 𝐴 ∧ βˆ€π‘§ ∈ suc π΄βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
953, 94eqtr4d 2776 1 (𝐴 ∈ On β†’ (cfβ€˜suc 𝐴) = 1o)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βˆ© cint 4951  dom cdm 5677  Oncon0 6365  suc csuc 6367  β€˜cfv 6544  1oc1o 8459  cardccrd 9930  cfccf 9932
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
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-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-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-cf 9936
This theorem is referenced by:  cflim2  10258  cfpwsdom  10579  rankcf  10772
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