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Theorem cflim2 9368
Description: The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
Hypothesis
Ref Expression
cflim2.1 𝐴 ∈ V
Assertion
Ref Expression
cflim2 (Lim 𝐴 ↔ Lim (cf‘𝐴))

Proof of Theorem cflim2
Dummy variables 𝑠 𝑦 𝑥 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabid 3302 . . . . . . 7 (𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} ↔ (𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴))
2 selpw 4356 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
3 limord 5995 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝐴 → Ord 𝐴)
4 ordsson 7217 . . . . . . . . . . . . . . . . . . . 20 (Ord 𝐴𝐴 ⊆ On)
5 sstr 3804 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
65expcom 400 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ On → (𝑦𝐴𝑦 ⊆ On))
73, 4, 63syl 18 . . . . . . . . . . . . . . . . . . 19 (Lim 𝐴 → (𝑦𝐴𝑦 ⊆ On))
87imp 395 . . . . . . . . . . . . . . . . . 18 ((Lim 𝐴𝑦𝐴) → 𝑦 ⊆ On)
983adant3 1155 . . . . . . . . . . . . . . . . 17 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → 𝑦 ⊆ On)
10 ssel2 3791 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ On ∧ 𝑠𝑦) → 𝑠 ∈ On)
11 eloni 5944 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ On → Ord 𝑠)
12 ordirr 5952 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑠 → ¬ 𝑠𝑠)
1310, 11, 123syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ On ∧ 𝑠𝑦) → ¬ 𝑠𝑠)
14 ssel 3790 . . . . . . . . . . . . . . . . . . . 20 (𝑦𝑠 → (𝑠𝑦𝑠𝑠))
1514com12 32 . . . . . . . . . . . . . . . . . . 19 (𝑠𝑦 → (𝑦𝑠𝑠𝑠))
1615adantl 469 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ On ∧ 𝑠𝑦) → (𝑦𝑠𝑠𝑠))
1713, 16mtod 189 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ On ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
189, 17sylan 571 . . . . . . . . . . . . . . . 16 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
19 simpl2 1237 . . . . . . . . . . . . . . . . 17 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → 𝑦𝐴)
20 sstr 3804 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴𝐴𝑠) → 𝑦𝑠)
2119, 20sylan 571 . . . . . . . . . . . . . . . 16 ((((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) ∧ 𝐴𝑠) → 𝑦𝑠)
2218, 21mtand 841 . . . . . . . . . . . . . . 15 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝐴𝑠)
23 simpl3 1239 . . . . . . . . . . . . . . . 16 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → 𝑦 = 𝐴)
2423sseq1d 3827 . . . . . . . . . . . . . . 15 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ( 𝑦𝑠𝐴𝑠))
2522, 24mtbird 316 . . . . . . . . . . . . . 14 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
26 unissb 4661 . . . . . . . . . . . . . 14 ( 𝑦𝑠 ↔ ∀𝑡𝑦 𝑡𝑠)
2725, 26sylnib 319 . . . . . . . . . . . . 13 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ ∀𝑡𝑦 𝑡𝑠)
2827nrexdv 3186 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ ∃𝑠𝑦𝑡𝑦 𝑡𝑠)
29 ssel 3790 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → (𝑠𝑦𝑠 ∈ On))
30 ssel 3790 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → (𝑡𝑦𝑡 ∈ On))
31 ontri1 5968 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ On ∧ 𝑠 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑠𝑡))
3231ancoms 448 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑠𝑡))
33 vex 3392 . . . . . . . . . . . . . . . . . . . . . 22 𝑡 ∈ V
34 vex 3392 . . . . . . . . . . . . . . . . . . . . . 22 𝑠 ∈ V
3533, 34brcnv 5504 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 E 𝑠𝑠 E 𝑡)
36 epel 5225 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 E 𝑡𝑠𝑡)
3735, 36bitri 266 . . . . . . . . . . . . . . . . . . . 20 (𝑡 E 𝑠𝑠𝑡)
3837notbii 311 . . . . . . . . . . . . . . . . . . 19 𝑡 E 𝑠 ↔ ¬ 𝑠𝑡)
3932, 38syl6bbr 280 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠))
4039a1i 11 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠)))
4129, 30, 40syl2and 597 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → ((𝑠𝑦𝑡𝑦) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠)))
4241impl 445 . . . . . . . . . . . . . . 15 (((𝑦 ⊆ On ∧ 𝑠𝑦) ∧ 𝑡𝑦) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠))
4342ralbidva 3171 . . . . . . . . . . . . . 14 ((𝑦 ⊆ On ∧ 𝑠𝑦) → (∀𝑡𝑦 𝑡𝑠 ↔ ∀𝑡𝑦 ¬ 𝑡 E 𝑠))
4443rexbidva 3235 . . . . . . . . . . . . 13 (𝑦 ⊆ On → (∃𝑠𝑦𝑡𝑦 𝑡𝑠 ↔ ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠))
459, 44syl 17 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → (∃𝑠𝑦𝑡𝑦 𝑡𝑠 ↔ ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠))
4628, 45mtbid 315 . . . . . . . . . . 11 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
47 vex 3392 . . . . . . . . . . . . 13 𝑦 ∈ V
4847a1i 11 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ V)
49 epweon 7211 . . . . . . . . . . . . . . . . . . 19 E We On
50 wess 5296 . . . . . . . . . . . . . . . . . . 19 (𝑦 ⊆ On → ( E We On → E We 𝑦))
5149, 50mpi 20 . . . . . . . . . . . . . . . . . 18 (𝑦 ⊆ On → E We 𝑦)
52 weso 5300 . . . . . . . . . . . . . . . . . 18 ( E We 𝑦 → E Or 𝑦)
5351, 52syl 17 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → E Or 𝑦)
54 cnvso 5886 . . . . . . . . . . . . . . . . 17 ( E Or 𝑦 E Or 𝑦)
5553, 54sylib 209 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → E Or 𝑦)
5655adantr 468 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → E Or 𝑦)
57 onssnum 9144 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
5847, 57mpan 673 . . . . . . . . . . . . . . . . . 18 (𝑦 ⊆ On → 𝑦 ∈ dom card)
59 cardid2 9060 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
60 ensym 8239 . . . . . . . . . . . . . . . . . 18 ((card‘𝑦) ≈ 𝑦𝑦 ≈ (card‘𝑦))
6158, 59, 603syl 18 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → 𝑦 ≈ (card‘𝑦))
62 nnsdom 8796 . . . . . . . . . . . . . . . . 17 ((card‘𝑦) ∈ ω → (card‘𝑦) ≺ ω)
63 ensdomtr 8333 . . . . . . . . . . . . . . . . 17 ((𝑦 ≈ (card‘𝑦) ∧ (card‘𝑦) ≺ ω) → 𝑦 ≺ ω)
6461, 62, 63syl2an 585 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ≺ ω)
65 isfinite 8794 . . . . . . . . . . . . . . . 16 (𝑦 ∈ Fin ↔ 𝑦 ≺ ω)
6664, 65sylibr 225 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
67 wofi 8446 . . . . . . . . . . . . . . 15 (( E Or 𝑦𝑦 ∈ Fin) → E We 𝑦)
6856, 66, 67syl2anc 575 . . . . . . . . . . . . . 14 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → E We 𝑦)
699, 68sylan 571 . . . . . . . . . . . . 13 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → E We 𝑦)
70 wefr 5299 . . . . . . . . . . . . 13 ( E We 𝑦 E Fr 𝑦)
7169, 70syl 17 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → E Fr 𝑦)
72 ssidd 3819 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦𝑦)
73 unieq 4636 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → 𝑦 = ∅)
74 uni0 4657 . . . . . . . . . . . . . . . . . . 19 ∅ = ∅
7573, 74syl6eq 2854 . . . . . . . . . . . . . . . . . 18 (𝑦 = ∅ → 𝑦 = ∅)
76 eqeq1 2808 . . . . . . . . . . . . . . . . . 18 ( 𝑦 = 𝐴 → ( 𝑦 = ∅ ↔ 𝐴 = ∅))
7775, 76syl5ib 235 . . . . . . . . . . . . . . . . 17 ( 𝑦 = 𝐴 → (𝑦 = ∅ → 𝐴 = ∅))
78 nlim0 5994 . . . . . . . . . . . . . . . . . 18 ¬ Lim ∅
79 limeq 5946 . . . . . . . . . . . . . . . . . 18 (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
8078, 79mtbiri 318 . . . . . . . . . . . . . . . . 17 (𝐴 = ∅ → ¬ Lim 𝐴)
8177, 80syl6 35 . . . . . . . . . . . . . . . 16 ( 𝑦 = 𝐴 → (𝑦 = ∅ → ¬ Lim 𝐴))
8281necon2ad 2991 . . . . . . . . . . . . . . 15 ( 𝑦 = 𝐴 → (Lim 𝐴𝑦 ≠ ∅))
8382impcom 396 . . . . . . . . . . . . . 14 ((Lim 𝐴 𝑦 = 𝐴) → 𝑦 ≠ ∅)
84833adant2 1154 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → 𝑦 ≠ ∅)
8584adantr 468 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ≠ ∅)
86 fri 5271 . . . . . . . . . . . 12 (((𝑦 ∈ V ∧ E Fr 𝑦) ∧ (𝑦𝑦𝑦 ≠ ∅)) → ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
8748, 71, 72, 85, 86syl22anc 858 . . . . . . . . . . 11 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
8846, 87mtand 841 . . . . . . . . . 10 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ (card‘𝑦) ∈ ω)
89 cardon 9051 . . . . . . . . . . 11 (card‘𝑦) ∈ On
90 eloni 5944 . . . . . . . . . . 11 ((card‘𝑦) ∈ On → Ord (card‘𝑦))
91 ordom 7302 . . . . . . . . . . . 12 Ord ω
92 ordtri1 5967 . . . . . . . . . . . 12 ((Ord ω ∧ Ord (card‘𝑦)) → (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω))
9391, 92mpan 673 . . . . . . . . . . 11 (Ord (card‘𝑦) → (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω))
9489, 90, 93mp2b 10 . . . . . . . . . 10 (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω)
9588, 94sylibr 225 . . . . . . . . 9 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ω ⊆ (card‘𝑦))
962, 95syl3an2b 1516 . . . . . . . 8 ((Lim 𝐴𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴) → ω ⊆ (card‘𝑦))
97963expb 1142 . . . . . . 7 ((Lim 𝐴 ∧ (𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴)) → ω ⊆ (card‘𝑦))
981, 97sylan2b 583 . . . . . 6 ((Lim 𝐴𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}) → ω ⊆ (card‘𝑦))
9998ralrimiva 3152 . . . . 5 (Lim 𝐴 → ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
100 ssiin 4760 . . . . 5 (ω ⊆ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦) ↔ ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
10199, 100sylibr 225 . . . 4 (Lim 𝐴 → ω ⊆ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦))
102 cflim2.1 . . . . 5 𝐴 ∈ V
103102cflim3 9367 . . . 4 (Lim 𝐴 → (cf‘𝐴) = 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦))
104101, 103sseqtr4d 3837 . . 3 (Lim 𝐴 → ω ⊆ (cf‘𝐴))
105 fvex 6419 . . . . . . 7 (card‘𝑦) ∈ V
106105dfiin2 4745 . . . . . 6 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦) = {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)}
107103, 106syl6eq 2854 . . . . 5 (Lim 𝐴 → (cf‘𝐴) = {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
108 cardlim 9079 . . . . . . . . 9 (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))
109 sseq2 3822 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ ω ⊆ (card‘𝑦)))
110 limeq 5946 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (Lim 𝑥 ↔ Lim (card‘𝑦)))
111109, 110bibi12d 336 . . . . . . . . 9 (𝑥 = (card‘𝑦) → ((ω ⊆ 𝑥 ↔ Lim 𝑥) ↔ (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))))
112108, 111mpbiri 249 . . . . . . . 8 (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
113112rexlimivw 3215 . . . . . . 7 (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
114113ss2abi 3869 . . . . . 6 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)}
115 eleq1 2871 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
11689, 115mpbiri 249 . . . . . . . . 9 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
117116rexlimivw 3215 . . . . . . . 8 (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦) → 𝑥 ∈ On)
118117abssi 3872 . . . . . . 7 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ On
119 fvex 6419 . . . . . . . . 9 (cf‘𝐴) ∈ V
120107, 119syl6eqelr 2892 . . . . . . . 8 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
121 intex 5010 . . . . . . . 8 ({𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅ ↔ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
122120, 121sylibr 225 . . . . . . 7 (Lim 𝐴 → {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅)
123 onint 7223 . . . . . . 7 (({𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ On ∧ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅) → {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
124118, 122, 123sylancr 577 . . . . . 6 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
125114, 124sseldi 3794 . . . . 5 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)})
126107, 125eqeltrd 2883 . . . 4 (Lim 𝐴 → (cf‘𝐴) ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)})
127 sseq2 3822 . . . . . 6 (𝑥 = (cf‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (cf‘𝐴)))
128 limeq 5946 . . . . . 6 (𝑥 = (cf‘𝐴) → (Lim 𝑥 ↔ Lim (cf‘𝐴)))
129127, 128bibi12d 336 . . . . 5 (𝑥 = (cf‘𝐴) → ((ω ⊆ 𝑥 ↔ Lim 𝑥) ↔ (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴))))
130119, 129elab 3543 . . . 4 ((cf‘𝐴) ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)} ↔ (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴)))
131126, 130sylib 209 . . 3 (Lim 𝐴 → (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴)))
132104, 131mpbid 223 . 2 (Lim 𝐴 → Lim (cf‘𝐴))
133 eloni 5944 . . . . . . 7 (𝐴 ∈ On → Ord 𝐴)
134 ordzsl 7273 . . . . . . 7 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
135133, 134sylib 209 . . . . . 6 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
136 df-3or 1101 . . . . . . 7 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ∨ Lim 𝐴))
137 orcom 888 . . . . . . 7 (((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ∨ Lim 𝐴) ↔ (Lim 𝐴 ∨ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
138 df-or 866 . . . . . . 7 ((Lim 𝐴 ∨ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
139136, 137, 1383bitri 288 . . . . . 6 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
140135, 139sylib 209 . . . . 5 (𝐴 ∈ On → (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
141 fveq2 6406 . . . . . . . . 9 (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
142 cf0 9356 . . . . . . . . 9 (cf‘∅) = ∅
143141, 142syl6eq 2854 . . . . . . . 8 (𝐴 = ∅ → (cf‘𝐴) = ∅)
144 limeq 5946 . . . . . . . 8 ((cf‘𝐴) = ∅ → (Lim (cf‘𝐴) ↔ Lim ∅))
145143, 144syl 17 . . . . . . 7 (𝐴 = ∅ → (Lim (cf‘𝐴) ↔ Lim ∅))
14678, 145mtbiri 318 . . . . . 6 (𝐴 = ∅ → ¬ Lim (cf‘𝐴))
147 1n0 7810 . . . . . . . . . 10 1𝑜 ≠ ∅
148 df1o2 7807 . . . . . . . . . . . 12 1𝑜 = {∅}
149148unieqi 4637 . . . . . . . . . . 11 1𝑜 = {∅}
150 0ex 4982 . . . . . . . . . . . 12 ∅ ∈ V
151150unisn 4644 . . . . . . . . . . 11 {∅} = ∅
152149, 151eqtri 2826 . . . . . . . . . 10 1𝑜 = ∅
153147, 152neeqtrri 3049 . . . . . . . . 9 1𝑜 1𝑜
154 limuni 5996 . . . . . . . . . 10 (Lim 1𝑜 → 1𝑜 = 1𝑜)
155154necon3ai 3001 . . . . . . . . 9 (1𝑜 1𝑜 → ¬ Lim 1𝑜)
156153, 155ax-mp 5 . . . . . . . 8 ¬ Lim 1𝑜
157 fveq2 6406 . . . . . . . . . 10 (𝐴 = suc 𝑥 → (cf‘𝐴) = (cf‘suc 𝑥))
158 cfsuc 9362 . . . . . . . . . 10 (𝑥 ∈ On → (cf‘suc 𝑥) = 1𝑜)
159157, 158sylan9eqr 2860 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (cf‘𝐴) = 1𝑜)
160 limeq 5946 . . . . . . . . 9 ((cf‘𝐴) = 1𝑜 → (Lim (cf‘𝐴) ↔ Lim 1𝑜))
161159, 160syl 17 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (Lim (cf‘𝐴) ↔ Lim 1𝑜))
162156, 161mtbiri 318 . . . . . . 7 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → ¬ Lim (cf‘𝐴))
163162rexlimiva 3214 . . . . . 6 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ Lim (cf‘𝐴))
164146, 163jaoi 875 . . . . 5 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → ¬ Lim (cf‘𝐴))
165140, 164syl6 35 . . . 4 (𝐴 ∈ On → (¬ Lim 𝐴 → ¬ Lim (cf‘𝐴)))
166165con4d 115 . . 3 (𝐴 ∈ On → (Lim (cf‘𝐴) → Lim 𝐴))
167 cff 9353 . . . . . . . . 9 cf:On⟶On
168167fdmi 6264 . . . . . . . 8 dom cf = On
169168eleq2i 2875 . . . . . . 7 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
170 ndmfv 6436 . . . . . . 7 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
171169, 170sylnbir 322 . . . . . 6 𝐴 ∈ On → (cf‘𝐴) = ∅)
172171, 144syl 17 . . . . 5 𝐴 ∈ On → (Lim (cf‘𝐴) ↔ Lim ∅))
17378, 172mtbiri 318 . . . 4 𝐴 ∈ On → ¬ Lim (cf‘𝐴))
174173pm2.21d 119 . . 3 𝐴 ∈ On → (Lim (cf‘𝐴) → Lim 𝐴))
175166, 174pm2.61i 176 . 2 (Lim (cf‘𝐴) → Lim 𝐴)
176132, 175impbii 200 1 (Lim 𝐴 ↔ Lim (cf‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3o 1099  w3a 1100   = wceq 1637  wcel 2156  {cab 2790  wne 2976  wral 3094  wrex 3095  {crab 3098  Vcvv 3389  wss 3767  c0 4114  𝒫 cpw 4349  {csn 4368   cuni 4628   cint 4667   ciin 4711   class class class wbr 4842   E cep 5221   Or wor 5229   Fr wfr 5265   We wwe 5267  ccnv 5308  dom cdm 5309  Ord word 5933  Oncon0 5934  Lim wlim 5935  suc csuc 5936  cfv 6099  ωcom 7293  1𝑜c1o 7787  cen 8187  csdm 8189  Fincfn 8190  cardccrd 9042  cfccf 9044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177  ax-inf2 8783
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ne 2977  df-ral 3099  df-rex 3100  df-reu 3101  df-rmo 3102  df-rab 3103  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4115  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-iin 4713  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5217  df-eprel 5222  df-po 5230  df-so 5231  df-fr 5268  df-se 5269  df-we 5270  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-rn 5320  df-res 5321  df-ima 5322  df-pred 5891  df-ord 5937  df-on 5938  df-lim 5939  df-suc 5940  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-isom 6108  df-riota 6833  df-om 7294  df-wrecs 7640  df-recs 7702  df-rdg 7740  df-1o 7794  df-er 7977  df-en 8191  df-dom 8192  df-sdom 8193  df-fin 8194  df-card 9046  df-cf 9048
This theorem is referenced by:  cfom  9369
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