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Theorem cfsuc 10297
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 onsucb 7837 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
2 cfval 10287 . . 3 (suc 𝐴 ∈ On → (cf‘suc 𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
31, 2sylbi 217 . 2 (𝐴 ∈ On → (cf‘suc 𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
4 cardsn 10009 . . . . . 6 (𝐴 ∈ On → (card‘{𝐴}) = 1o)
54eqcomd 2743 . . . . 5 (𝐴 ∈ On → 1o = (card‘{𝐴}))
6 snidg 4660 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ {𝐴})
7 elsuci 6451 . . . . . . . . 9 (𝑧 ∈ suc 𝐴 → (𝑧𝐴𝑧 = 𝐴))
8 onelss 6426 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧𝐴𝑧𝐴))
9 eqimss 4042 . . . . . . . . . . 11 (𝑧 = 𝐴𝑧𝐴)
109a1i 11 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧 = 𝐴𝑧𝐴))
118, 10jaod 860 . . . . . . . . 9 (𝐴 ∈ On → ((𝑧𝐴𝑧 = 𝐴) → 𝑧𝐴))
127, 11syl5 34 . . . . . . . 8 (𝐴 ∈ On → (𝑧 ∈ suc 𝐴𝑧𝐴))
13 sseq2 4010 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑧𝑤𝑧𝐴))
1413rspcev 3622 . . . . . . . 8 ((𝐴 ∈ {𝐴} ∧ 𝑧𝐴) → ∃𝑤 ∈ {𝐴}𝑧𝑤)
156, 12, 14syl6an 684 . . . . . . 7 (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → ∃𝑤 ∈ {𝐴}𝑧𝑤))
1615ralrimiv 3145 . . . . . 6 (𝐴 ∈ On → ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)
17 ssun2 4179 . . . . . . 7 {𝐴} ⊆ (𝐴 ∪ {𝐴})
18 df-suc 6390 . . . . . . 7 suc 𝐴 = (𝐴 ∪ {𝐴})
1917, 18sseqtrri 4033 . . . . . 6 {𝐴} ⊆ suc 𝐴
2016, 19jctil 519 . . . . 5 (𝐴 ∈ On → ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))
21 snex 5436 . . . . . 6 {𝐴} ∈ V
22 fveq2 6906 . . . . . . . 8 (𝑦 = {𝐴} → (card‘𝑦) = (card‘{𝐴}))
2322eqeq2d 2748 . . . . . . 7 (𝑦 = {𝐴} → (1o = (card‘𝑦) ↔ 1o = (card‘{𝐴})))
24 sseq1 4009 . . . . . . . 8 (𝑦 = {𝐴} → (𝑦 ⊆ suc 𝐴 ↔ {𝐴} ⊆ suc 𝐴))
25 rexeq 3322 . . . . . . . . 9 (𝑦 = {𝐴} → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ {𝐴}𝑧𝑤))
2625ralbidv 3178 . . . . . . . 8 (𝑦 = {𝐴} → (∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))
2724, 26anbi12d 632 . . . . . . 7 (𝑦 = {𝐴} → ((𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤) ↔ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)))
2823, 27anbi12d 632 . . . . . 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 3606 . . . . 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 584 . . . 4 (𝐴 ∈ On → ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
31 1oex 8516 . . . . 5 1o ∈ V
32 eqeq1 2741 . . . . . . 7 (𝑥 = 1o → (𝑥 = (card‘𝑦) ↔ 1o = (card‘𝑦)))
3332anbi1d 631 . . . . . 6 (𝑥 = 1o → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
3433exbidv 1921 . . . . 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 3679 . . . 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 234 . . 3 (𝐴 ∈ On → 1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
37 el1o 8533 . . . . 5 (𝑣 ∈ 1o𝑣 = ∅)
38 eqcom 2744 . . . . . . . . . . . . . . 15 (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
39 vex 3484 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
40 onssnum 10080 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
4139, 40mpan 690 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → 𝑦 ∈ dom card)
42 cardnueq0 10004 . . . . . . . . . . . . . . . 16 (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4341, 42syl 17 . . . . . . . . . . . . . . 15 (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4438, 43bitrid 283 . . . . . . . . . . . . . 14 (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
4544biimpa 476 . . . . . . . . . . . . 13 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
46 rex0 4360 . . . . . . . . . . . . . . . . 17 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
4746a1i 11 . . . . . . . . . . . . . . . 16 (𝑧 ∈ suc 𝐴 → ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
4847nrex 3074 . . . . . . . . . . . . . . 15 ¬ ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤
49 nsuceq0 6467 . . . . . . . . . . . . . . . 16 suc 𝐴 ≠ ∅
50 r19.2z 4495 . . . . . . . . . . . . . . . 16 ((suc 𝐴 ≠ ∅ ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤) → ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤)
5149, 50mpan 690 . . . . . . . . . . . . . . 15 (∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤 → ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤)
5248, 51mto 197 . . . . . . . . . . . . . 14 ¬ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤
53 rexeq 3322 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧𝑤))
5453ralbidv 3178 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤))
5552, 54mtbiri 327 . . . . . . . . . . . . 13 (𝑦 = ∅ → ¬ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)
5645, 55syl 17 . . . . . . . . . . . 12 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)
5756intnand 488 . . . . . . . . . . 11 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
58 imnan 399 . . . . . . . . . . 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 230 . . . . . . . . . 10 ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
60 onsuc 7831 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → suc 𝐴 ∈ On)
61 onss 7805 . . . . . . . . . . . . . . . . 17 (suc 𝐴 ∈ On → suc 𝐴 ⊆ On)
62 sstr 3992 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ⊆ On) → 𝑦 ⊆ On)
6361, 62sylan2 593 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ∈ On) → 𝑦 ⊆ On)
6460, 63sylan2 593 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ suc 𝐴𝐴 ∈ On) → 𝑦 ⊆ On)
6564ancoms 458 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ⊆ suc 𝐴) → 𝑦 ⊆ On)
6665adantrr 717 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
67663adant2 1132 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
68 simp2 1138 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ∅ = (card‘𝑦))
69 simp3 1139 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
7067, 68, 69jca31 514 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
71703expib 1123 . . . . . . . . . 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 199 . . . . . . . . 9 (𝐴 ∈ On → ¬ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
7372nexdv 1936 . . . . . . . 8 (𝐴 ∈ On → ¬ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
74 0ex 5307 . . . . . . . . 9 ∅ ∈ V
75 eqeq1 2741 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
7675anbi1d 631 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7776exbidv 1921 . . . . . . . . 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 3679 . . . . . . . 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 329 . . . . . . 7 (𝐴 ∈ On → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8079adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
81 eleq1 2829 . . . . . . 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 481 . . . . . 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 325 . . . . 5 ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8437, 83sylan2b 594 . . . 4 ((𝐴 ∈ On ∧ 𝑣 ∈ 1o) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8584ralrimiva 3146 . . 3 (𝐴 ∈ On → ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
86 cardon 9984 . . . . . . . 8 (card‘𝑦) ∈ On
87 eleq1 2829 . . . . . . . 8 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
8886, 87mpbiri 258 . . . . . . 7 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
8988adantr 480 . . . . . 6 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑥 ∈ On)
9089exlimiv 1930 . . . . 5 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑥 ∈ On)
9190abssi 4070 . . . 4 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
92 oneqmini 6436 . . . 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 584 . 2 (𝐴 ∈ On → 1o = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
953, 94eqtr4d 2780 1 (𝐴 ∈ On → (cf‘suc 𝐴) = 1o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cun 3949  wss 3951  c0 4333  {csn 4626   cint 4946  dom cdm 5685  Oncon0 6384  suc csuc 6386  cfv 6561  1oc1o 8499  cardccrd 9975  cfccf 9977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-cf 9981
This theorem is referenced by:  cflim2  10303  cfpwsdom  10624  rankcf  10817
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