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Theorem cfsuc 9117
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 7059 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
2 cfval 9107 . . 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 8833 . . . . . 6 (𝐴 ∈ On → (card‘{𝐴}) = 1𝑜)
54eqcomd 2657 . . . . 5 (𝐴 ∈ On → 1𝑜 = (card‘{𝐴}))
6 snidg 4239 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ {𝐴})
7 elsuci 5829 . . . . . . . . 9 (𝑧 ∈ suc 𝐴 → (𝑧𝐴𝑧 = 𝐴))
8 onelss 5804 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧𝐴𝑧𝐴))
9 eqimss 3690 . . . . . . . . . . 11 (𝑧 = 𝐴𝑧𝐴)
109a1i 11 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧 = 𝐴𝑧𝐴))
118, 10jaod 394 . . . . . . . . 9 (𝐴 ∈ On → ((𝑧𝐴𝑧 = 𝐴) → 𝑧𝐴))
127, 11syl5 34 . . . . . . . 8 (𝐴 ∈ On → (𝑧 ∈ suc 𝐴𝑧𝐴))
13 sseq2 3660 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑧𝑤𝑧𝐴))
1413rspcev 3340 . . . . . . . 8 ((𝐴 ∈ {𝐴} ∧ 𝑧𝐴) → ∃𝑤 ∈ {𝐴}𝑧𝑤)
156, 12, 14syl6an 567 . . . . . . 7 (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → ∃𝑤 ∈ {𝐴}𝑧𝑤))
1615ralrimiv 2994 . . . . . 6 (𝐴 ∈ On → ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)
17 ssun2 3810 . . . . . . 7 {𝐴} ⊆ (𝐴 ∪ {𝐴})
18 df-suc 5767 . . . . . . 7 suc 𝐴 = (𝐴 ∪ {𝐴})
1917, 18sseqtr4i 3671 . . . . . 6 {𝐴} ⊆ suc 𝐴
2016, 19jctil 559 . . . . 5 (𝐴 ∈ On → ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))
21 snex 4938 . . . . . 6 {𝐴} ∈ V
22 fveq2 6229 . . . . . . . 8 (𝑦 = {𝐴} → (card‘𝑦) = (card‘{𝐴}))
2322eqeq2d 2661 . . . . . . 7 (𝑦 = {𝐴} → (1𝑜 = (card‘𝑦) ↔ 1𝑜 = (card‘{𝐴})))
24 sseq1 3659 . . . . . . . 8 (𝑦 = {𝐴} → (𝑦 ⊆ suc 𝐴 ↔ {𝐴} ⊆ suc 𝐴))
25 rexeq 3169 . . . . . . . . 9 (𝑦 = {𝐴} → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ {𝐴}𝑧𝑤))
2625ralbidv 3015 . . . . . . . 8 (𝑦 = {𝐴} → (∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))
2724, 26anbi12d 747 . . . . . . 7 (𝑦 = {𝐴} → ((𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤) ↔ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)))
2823, 27anbi12d 747 . . . . . 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 3331 . . . . 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 694 . . . 4 (𝐴 ∈ On → ∃𝑦(1𝑜 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
31 1on 7612 . . . . . 6 1𝑜 ∈ On
3231elexi 3244 . . . . 5 1𝑜 ∈ V
33 eqeq1 2655 . . . . . . 7 (𝑥 = 1𝑜 → (𝑥 = (card‘𝑦) ↔ 1𝑜 = (card‘𝑦)))
3433anbi1d 741 . . . . . 6 (𝑥 = 1𝑜 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (1𝑜 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
3534exbidv 1890 . . . . 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 3382 . . . 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 7624 . . . . 5 (𝑣 ∈ 1𝑜𝑣 = ∅)
39 eqcom 2658 . . . . . . . . . . . . . . 15 (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
40 vex 3234 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
41 onssnum 8901 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
4240, 41mpan 706 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → 𝑦 ∈ dom card)
43 cardnueq0 8828 . . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . 13 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
47 rex0 3971 . . . . . . . . . . . . . . . . 17 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
4847a1i 11 . . . . . . . . . . . . . . . 16 (𝑧 ∈ suc 𝐴 → ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
4948nrex 3029 . . . . . . . . . . . . . . 15 ¬ ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤
50 nsuceq0 5843 . . . . . . . . . . . . . . . 16 suc 𝐴 ≠ ∅
51 r19.2z 4093 . . . . . . . . . . . . . . . 16 ((suc 𝐴 ≠ ∅ ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤) → ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤)
5250, 51mpan 706 . . . . . . . . . . . . . . 15 (∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤 → ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤)
5349, 52mto 188 . . . . . . . . . . . . . 14 ¬ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤
54 rexeq 3169 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧𝑤))
5554ralbidv 3015 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤))
5653, 55mtbiri 316 . . . . . . . . . . . . 13 (𝑦 = ∅ → ¬ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)
5746, 56syl 17 . . . . . . . . . . . 12 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)
5857intnand 982 . . . . . . . . . . 11 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
59 imnan 437 . . . . . . . . . . 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 7055 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → suc 𝐴 ∈ On)
62 onss 7032 . . . . . . . . . . . . . . . . 17 (suc 𝐴 ∈ On → suc 𝐴 ⊆ On)
63 sstr 3644 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ⊆ On) → 𝑦 ⊆ On)
6462, 63sylan2 490 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ∈ On) → 𝑦 ⊆ On)
6561, 64sylan2 490 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ suc 𝐴𝐴 ∈ On) → 𝑦 ⊆ On)
6665ancoms 468 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ⊆ suc 𝐴) → 𝑦 ⊆ On)
6766adantrr 753 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
68673adant2 1100 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
69 simp2 1082 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ∅ = (card‘𝑦))
70 simp3 1083 . . . . . . . . . . . 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 1287 . . . . . . . . . 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 1904 . . . . . . . 8 (𝐴 ∈ On → ¬ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
75 0ex 4823 . . . . . . . . 9 ∅ ∈ V
76 eqeq1 2655 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
7776anbi1d 741 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7877exbidv 1890 . . . . . . . . 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 3382 . . . . . . . 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 318 . . . . . . 7 (𝐴 ∈ On → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8180adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
82 eleq1 2718 . . . . . . 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 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))}))
8481, 83mtbird 314 . . . . 5 ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8538, 84sylan2b 491 . . . 4 ((𝐴 ∈ On ∧ 𝑣 ∈ 1𝑜) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8685ralrimiva 2995 . . 3 (𝐴 ∈ On → ∀𝑣 ∈ 1𝑜 ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
87 cardon 8808 . . . . . . . 8 (card‘𝑦) ∈ On
88 eleq1 2718 . . . . . . . 8 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
8987, 88mpbiri 248 . . . . . . 7 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
9089adantr 480 . . . . . 6 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑥 ∈ On)
9190exlimiv 1898 . . . . 5 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑥 ∈ On)
9291abssi 3710 . . . 4 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
93 oneqmini 5814 . . . 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 694 . 2 (𝐴 ∈ On → 1𝑜 = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
963, 95eqtr4d 2688 1 (𝐴 ∈ On → (cf‘suc 𝐴) = 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cun 3605  wss 3607  c0 3948  {csn 4210   cint 4507  dom cdm 5143  Oncon0 5761  suc csuc 5763  cfv 5926  1𝑜c1o 7598  cardccrd 8799  cfccf 8801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cf 8805
This theorem is referenced by:  cflim2  9123  cfpwsdom  9444  rankcf  9637
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