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Theorem cfsuc 9026
 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 6967 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
2 cfval 9016 . . 3 (suc 𝐴 ∈ On → (cf‘suc 𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
31, 2sylbi 207 . 2 (𝐴 ∈ On → (cf‘suc 𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
4 cardsn 8742 . . . . . 6 (𝐴 ∈ On → (card‘{𝐴}) = 1𝑜)
54eqcomd 2627 . . . . 5 (𝐴 ∈ On → 1𝑜 = (card‘{𝐴}))
6 snidg 4179 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ {𝐴})
7 elsuci 5752 . . . . . . . . 9 (𝑧 ∈ suc 𝐴 → (𝑧𝐴𝑧 = 𝐴))
8 onelss 5727 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧𝐴𝑧𝐴))
9 eqimss 3638 . . . . . . . . . . 11 (𝑧 = 𝐴𝑧𝐴)
109a1i 11 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧 = 𝐴𝑧𝐴))
118, 10jaod 395 . . . . . . . . 9 (𝐴 ∈ On → ((𝑧𝐴𝑧 = 𝐴) → 𝑧𝐴))
127, 11syl5 34 . . . . . . . 8 (𝐴 ∈ On → (𝑧 ∈ suc 𝐴𝑧𝐴))
13 sseq2 3608 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑧𝑤𝑧𝐴))
1413rspcev 3295 . . . . . . . 8 ((𝐴 ∈ {𝐴} ∧ 𝑧𝐴) → ∃𝑤 ∈ {𝐴}𝑧𝑤)
156, 12, 14syl6an 567 . . . . . . 7 (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → ∃𝑤 ∈ {𝐴}𝑧𝑤))
1615ralrimiv 2959 . . . . . 6 (𝐴 ∈ On → ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)
17 ssun2 3757 . . . . . . 7 {𝐴} ⊆ (𝐴 ∪ {𝐴})
18 df-suc 5690 . . . . . . 7 suc 𝐴 = (𝐴 ∪ {𝐴})
1917, 18sseqtr4i 3619 . . . . . 6 {𝐴} ⊆ suc 𝐴
2016, 19jctil 559 . . . . 5 (𝐴 ∈ On → ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))
21 snex 4871 . . . . . 6 {𝐴} ∈ V
22 fveq2 6150 . . . . . . . 8 (𝑦 = {𝐴} → (card‘𝑦) = (card‘{𝐴}))
2322eqeq2d 2631 . . . . . . 7 (𝑦 = {𝐴} → (1𝑜 = (card‘𝑦) ↔ 1𝑜 = (card‘{𝐴})))
24 sseq1 3607 . . . . . . . 8 (𝑦 = {𝐴} → (𝑦 ⊆ suc 𝐴 ↔ {𝐴} ⊆ suc 𝐴))
25 rexeq 3128 . . . . . . . . 9 (𝑦 = {𝐴} → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ {𝐴}𝑧𝑤))
2625ralbidv 2980 . . . . . . . 8 (𝑦 = {𝐴} → (∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))
2724, 26anbi12d 746 . . . . . . 7 (𝑦 = {𝐴} → ((𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤) ↔ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)))
2823, 27anbi12d 746 . . . . . 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 3286 . . . . 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 692 . . . 4 (𝐴 ∈ On → ∃𝑦(1𝑜 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
31 1on 7515 . . . . . 6 1𝑜 ∈ On
3231elexi 3199 . . . . 5 1𝑜 ∈ V
33 eqeq1 2625 . . . . . . 7 (𝑥 = 1𝑜 → (𝑥 = (card‘𝑦) ↔ 1𝑜 = (card‘𝑦)))
3433anbi1d 740 . . . . . 6 (𝑥 = 1𝑜 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (1𝑜 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
3534exbidv 1847 . . . . 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))))
3632, 35elab 3334 . . . 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)))
3730, 36sylibr 224 . . 3 (𝐴 ∈ On → 1𝑜 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
38 el1o 7527 . . . . 5 (𝑣 ∈ 1𝑜𝑣 = ∅)
39 eqcom 2628 . . . . . . . . . . . . . . 15 (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
40 vex 3189 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
41 onssnum 8810 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
4240, 41mpan 705 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → 𝑦 ∈ dom card)
43 cardnueq0 8737 . . . . . . . . . . . . . . . 16 (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4442, 43syl 17 . . . . . . . . . . . . . . 15 (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4539, 44syl5bb 272 . . . . . . . . . . . . . 14 (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
4645biimpa 501 . . . . . . . . . . . . 13 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
47 rex0 3916 . . . . . . . . . . . . . . . . 17 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
4847a1i 11 . . . . . . . . . . . . . . . 16 (𝑧 ∈ suc 𝐴 → ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
4948nrex 2994 . . . . . . . . . . . . . . 15 ¬ ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤
50 nsuceq0 5766 . . . . . . . . . . . . . . . 16 suc 𝐴 ≠ ∅
51 r19.2z 4034 . . . . . . . . . . . . . . . 16 ((suc 𝐴 ≠ ∅ ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤) → ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤)
5250, 51mpan 705 . . . . . . . . . . . . . . 15 (∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤 → ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤)
5349, 52mto 188 . . . . . . . . . . . . . 14 ¬ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤
54 rexeq 3128 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧𝑤))
5554ralbidv 2980 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤))
5653, 55mtbiri 317 . . . . . . . . . . . . 13 (𝑦 = ∅ → ¬ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)
5746, 56syl 17 . . . . . . . . . . . 12 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)
5857intnand 961 . . . . . . . . . . 11 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
59 imnan 438 . . . . . . . . . . 11 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
6058, 59mpbi 220 . . . . . . . . . 10 ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
61 suceloni 6963 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → suc 𝐴 ∈ On)
62 onss 6940 . . . . . . . . . . . . . . . . 17 (suc 𝐴 ∈ On → suc 𝐴 ⊆ On)
63 sstr 3592 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ⊆ On) → 𝑦 ⊆ On)
6462, 63sylan2 491 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ∈ On) → 𝑦 ⊆ On)
6561, 64sylan2 491 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ suc 𝐴𝐴 ∈ On) → 𝑦 ⊆ On)
6665ancoms 469 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ⊆ suc 𝐴) → 𝑦 ⊆ On)
6766adantrr 752 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
68673adant2 1078 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
69 simp2 1060 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ∅ = (card‘𝑦))
70 simp3 1061 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
7168, 69, 70jca31 556 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
72713expib 1265 . . . . . . . . . 10 (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7360, 72mtoi 190 . . . . . . . . 9 (𝐴 ∈ On → ¬ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
7473nexdv 1861 . . . . . . . 8 (𝐴 ∈ On → ¬ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
75 0ex 4752 . . . . . . . . 9 ∅ ∈ V
76 eqeq1 2625 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
7776anbi1d 740 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7877exbidv 1847 . . . . . . . . 9 (𝑥 = ∅ → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7975, 78elab 3334 . . . . . . . 8 (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
8074, 79sylnibr 319 . . . . . . 7 (𝐴 ∈ On → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8180adantr 481 . . . . . 6 ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
82 eleq1 2686 . . . . . . 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))}))
8382adantl 482 . . . . . 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))}))
8481, 83mtbird 315 . . . . 5 ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8538, 84sylan2b 492 . . . 4 ((𝐴 ∈ On ∧ 𝑣 ∈ 1𝑜) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8685ralrimiva 2960 . . 3 (𝐴 ∈ On → ∀𝑣 ∈ 1𝑜 ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
87 cardon 8717 . . . . . . . 8 (card‘𝑦) ∈ On
88 eleq1 2686 . . . . . . . 8 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
8987, 88mpbiri 248 . . . . . . 7 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
9089adantr 481 . . . . . 6 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑥 ∈ On)
9190exlimiv 1855 . . . . 5 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑥 ∈ On)
9291abssi 3658 . . . 4 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
93 oneqmini 5737 . . . 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))}))
9492, 93ax-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))})
9537, 86, 94syl2anc 692 . 2 (𝐴 ∈ On → 1𝑜 = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
963, 95eqtr4d 2658 1 (𝐴 ∈ On → (cf‘suc 𝐴) = 1𝑜)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  {cab 2607   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  Vcvv 3186   ∪ cun 3554   ⊆ wss 3556  ∅c0 3893  {csn 4150  ∩ cint 4442  dom cdm 5076  Oncon0 5684  suc csuc 5686  ‘cfv 5849  1𝑜c1o 7501  cardccrd 8708  cfccf 8710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-om 7016  df-wrecs 7355  df-recs 7416  df-1o 7508  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-card 8712  df-cf 8714 This theorem is referenced by:  cflim2  9032  cfpwsdom  9353  rankcf  9546
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