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Theorem cfsuc 9670
 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 → (cf‘suc 𝐴) = 1o)

Proof of Theorem cfsuc
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sucelon 7514 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
2 cfval 9660 . . 3 (suc 𝐴 ∈ On → (cf‘suc 𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
31, 2sylbi 220 . 2 (𝐴 ∈ On → (cf‘suc 𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
4 cardsn 9384 . . . . . 6 (𝐴 ∈ On → (card‘{𝐴}) = 1o)
54eqcomd 2804 . . . . 5 (𝐴 ∈ On → 1o = (card‘{𝐴}))
6 snidg 4559 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ {𝐴})
7 elsuci 6225 . . . . . . . . 9 (𝑧 ∈ suc 𝐴 → (𝑧𝐴𝑧 = 𝐴))
8 onelss 6201 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧𝐴𝑧𝐴))
9 eqimss 3971 . . . . . . . . . . 11 (𝑧 = 𝐴𝑧𝐴)
109a1i 11 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧 = 𝐴𝑧𝐴))
118, 10jaod 856 . . . . . . . . 9 (𝐴 ∈ On → ((𝑧𝐴𝑧 = 𝐴) → 𝑧𝐴))
127, 11syl5 34 . . . . . . . 8 (𝐴 ∈ On → (𝑧 ∈ suc 𝐴𝑧𝐴))
13 sseq2 3941 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑧𝑤𝑧𝐴))
1413rspcev 3571 . . . . . . . 8 ((𝐴 ∈ {𝐴} ∧ 𝑧𝐴) → ∃𝑤 ∈ {𝐴}𝑧𝑤)
156, 12, 14syl6an 683 . . . . . . 7 (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → ∃𝑤 ∈ {𝐴}𝑧𝑤))
1615ralrimiv 3148 . . . . . 6 (𝐴 ∈ On → ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)
17 ssun2 4100 . . . . . . 7 {𝐴} ⊆ (𝐴 ∪ {𝐴})
18 df-suc 6165 . . . . . . 7 suc 𝐴 = (𝐴 ∪ {𝐴})
1917, 18sseqtrri 3952 . . . . . 6 {𝐴} ⊆ suc 𝐴
2016, 19jctil 523 . . . . 5 (𝐴 ∈ On → ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))
21 snex 5297 . . . . . 6 {𝐴} ∈ V
22 fveq2 6645 . . . . . . . 8 (𝑦 = {𝐴} → (card‘𝑦) = (card‘{𝐴}))
2322eqeq2d 2809 . . . . . . 7 (𝑦 = {𝐴} → (1o = (card‘𝑦) ↔ 1o = (card‘{𝐴})))
24 sseq1 3940 . . . . . . . 8 (𝑦 = {𝐴} → (𝑦 ⊆ suc 𝐴 ↔ {𝐴} ⊆ suc 𝐴))
25 rexeq 3359 . . . . . . . . 9 (𝑦 = {𝐴} → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ {𝐴}𝑧𝑤))
2625ralbidv 3162 . . . . . . . 8 (𝑦 = {𝐴} → (∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))
2724, 26anbi12d 633 . . . . . . 7 (𝑦 = {𝐴} → ((𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤) ↔ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)))
2823, 27anbi12d 633 . . . . . 6 (𝑦 = {𝐴} → ((1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (1o = (card‘{𝐴}) ∧ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))))
2921, 28spcev 3555 . . . . 5 ((1o = (card‘{𝐴}) ∧ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)) → ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
305, 20, 29syl2anc 587 . . . 4 (𝐴 ∈ On → ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
31 1oex 8095 . . . . 5 1o ∈ V
32 eqeq1 2802 . . . . . . 7 (𝑥 = 1o → (𝑥 = (card‘𝑦) ↔ 1o = (card‘𝑦)))
3332anbi1d 632 . . . . . 6 (𝑥 = 1o → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
3433exbidv 1922 . . . . 5 (𝑥 = 1o → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
3531, 34elab 3615 . . . 4 (1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
3630, 35sylibr 237 . . 3 (𝐴 ∈ On → 1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
37 el1o 8109 . . . . 5 (𝑣 ∈ 1o𝑣 = ∅)
38 eqcom 2805 . . . . . . . . . . . . . . 15 (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
39 vex 3444 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
40 onssnum 9453 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
4139, 40mpan 689 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → 𝑦 ∈ dom card)
42 cardnueq0 9379 . . . . . . . . . . . . . . . 16 (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4341, 42syl 17 . . . . . . . . . . . . . . 15 (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4438, 43syl5bb 286 . . . . . . . . . . . . . 14 (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
4544biimpa 480 . . . . . . . . . . . . 13 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
46 rex0 4271 . . . . . . . . . . . . . . . . 17 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
4746a1i 11 . . . . . . . . . . . . . . . 16 (𝑧 ∈ suc 𝐴 → ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
4847nrex 3228 . . . . . . . . . . . . . . 15 ¬ ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤
49 nsuceq0 6239 . . . . . . . . . . . . . . . 16 suc 𝐴 ≠ ∅
50 r19.2z 4398 . . . . . . . . . . . . . . . 16 ((suc 𝐴 ≠ ∅ ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤) → ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤)
5149, 50mpan 689 . . . . . . . . . . . . . . 15 (∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤 → ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤)
5248, 51mto 200 . . . . . . . . . . . . . 14 ¬ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤
53 rexeq 3359 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧𝑤))
5453ralbidv 3162 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤))
5552, 54mtbiri 330 . . . . . . . . . . . . 13 (𝑦 = ∅ → ¬ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)
5645, 55syl 17 . . . . . . . . . . . 12 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)
5756intnand 492 . . . . . . . . . . 11 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
58 imnan 403 . . . . . . . . . . 11 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
5957, 58mpbi 233 . . . . . . . . . 10 ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
60 suceloni 7510 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → suc 𝐴 ∈ On)
61 onss 7487 . . . . . . . . . . . . . . . . 17 (suc 𝐴 ∈ On → suc 𝐴 ⊆ On)
62 sstr 3923 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ⊆ On) → 𝑦 ⊆ On)
6361, 62sylan2 595 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ∈ On) → 𝑦 ⊆ On)
6460, 63sylan2 595 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ suc 𝐴𝐴 ∈ On) → 𝑦 ⊆ On)
6564ancoms 462 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ⊆ suc 𝐴) → 𝑦 ⊆ On)
6665adantrr 716 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
67663adant2 1128 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
68 simp2 1134 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ∅ = (card‘𝑦))
69 simp3 1135 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
7067, 68, 69jca31 518 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
71703expib 1119 . . . . . . . . . 10 (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7259, 71mtoi 202 . . . . . . . . 9 (𝐴 ∈ On → ¬ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
7372nexdv 1937 . . . . . . . 8 (𝐴 ∈ On → ¬ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
74 0ex 5175 . . . . . . . . 9 ∅ ∈ V
75 eqeq1 2802 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
7675anbi1d 632 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7776exbidv 1922 . . . . . . . . 9 (𝑥 = ∅ → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7874, 77elab 3615 . . . . . . . 8 (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
7973, 78sylnibr 332 . . . . . . 7 (𝐴 ∈ On → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8079adantr 484 . . . . . 6 ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
81 eleq1 2877 . . . . . . 7 (𝑣 = ∅ → (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))}))
8281adantl 485 . . . . . 6 ((𝐴 ∈ On ∧ 𝑣 = ∅) → (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))}))
8380, 82mtbird 328 . . . . 5 ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8437, 83sylan2b 596 . . . 4 ((𝐴 ∈ On ∧ 𝑣 ∈ 1o) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8584ralrimiva 3149 . . 3 (𝐴 ∈ On → ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
86 cardon 9359 . . . . . . . 8 (card‘𝑦) ∈ On
87 eleq1 2877 . . . . . . . 8 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
8886, 87mpbiri 261 . . . . . . 7 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
8988adantr 484 . . . . . 6 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑥 ∈ On)
9089exlimiv 1931 . . . . 5 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑥 ∈ On)
9190abssi 3997 . . . 4 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
92 oneqmini 6210 . . . 4 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ⊆ On → ((1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ∧ ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))}) → 1o = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))}))
9391, 92ax-mp 5 . . 3 ((1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ∧ ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))}) → 1o = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
9436, 85, 93syl2anc 587 . 2 (𝐴 ∈ On → 1o = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
953, 94eqtr4d 2836 1 (𝐴 ∈ On → (cf‘suc 𝐴) = 1o)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  ∅c0 4243  {csn 4525  ∩ cint 4838  dom cdm 5519  Oncon0 6159  suc csuc 6161  ‘cfv 6324  1oc1o 8080  cardccrd 9350  cfccf 9352 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-om 7563  df-wrecs 7932  df-recs 7993  df-1o 8087  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-card 9354  df-cf 9356 This theorem is referenced by:  cflim2  9676  cfpwsdom  9997  rankcf  10190
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