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Theorem cflim2 9683
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 3369 . . . . . . 7 (𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} ↔ (𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴))
2 velpw 4527 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
3 limord 6237 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝐴 → Ord 𝐴)
4 ordsson 7498 . . . . . . . . . . . . . . . . . . . 20 (Ord 𝐴𝐴 ⊆ On)
5 sstr 3961 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
65expcom 417 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ On → (𝑦𝐴𝑦 ⊆ On))
73, 4, 63syl 18 . . . . . . . . . . . . . . . . . . 19 (Lim 𝐴 → (𝑦𝐴𝑦 ⊆ On))
87imp 410 . . . . . . . . . . . . . . . . . 18 ((Lim 𝐴𝑦𝐴) → 𝑦 ⊆ On)
983adant3 1129 . . . . . . . . . . . . . . . . 17 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → 𝑦 ⊆ On)
10 ssel2 3948 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ On ∧ 𝑠𝑦) → 𝑠 ∈ On)
11 eloni 6188 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ On → Ord 𝑠)
12 ordirr 6196 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑠 → ¬ 𝑠𝑠)
1310, 11, 123syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ On ∧ 𝑠𝑦) → ¬ 𝑠𝑠)
14 ssel 3946 . . . . . . . . . . . . . . . . . . . 20 (𝑦𝑠 → (𝑠𝑦𝑠𝑠))
1514com12 32 . . . . . . . . . . . . . . . . . . 19 (𝑠𝑦 → (𝑦𝑠𝑠𝑠))
1615adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ On ∧ 𝑠𝑦) → (𝑦𝑠𝑠𝑠))
1713, 16mtod 201 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ On ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
189, 17sylan 583 . . . . . . . . . . . . . . . 16 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
19 simpl2 1189 . . . . . . . . . . . . . . . . 17 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → 𝑦𝐴)
20 sstr 3961 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴𝐴𝑠) → 𝑦𝑠)
2119, 20sylan 583 . . . . . . . . . . . . . . . 16 ((((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) ∧ 𝐴𝑠) → 𝑦𝑠)
2218, 21mtand 815 . . . . . . . . . . . . . . 15 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝐴𝑠)
23 simpl3 1190 . . . . . . . . . . . . . . . 16 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → 𝑦 = 𝐴)
2423sseq1d 3984 . . . . . . . . . . . . . . 15 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ( 𝑦𝑠𝐴𝑠))
2522, 24mtbird 328 . . . . . . . . . . . . . 14 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
26 unissb 4856 . . . . . . . . . . . . . 14 ( 𝑦𝑠 ↔ ∀𝑡𝑦 𝑡𝑠)
2725, 26sylnib 331 . . . . . . . . . . . . 13 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ ∀𝑡𝑦 𝑡𝑠)
2827nrexdv 3262 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ ∃𝑠𝑦𝑡𝑦 𝑡𝑠)
29 ssel 3946 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → (𝑠𝑦𝑠 ∈ On))
30 ssel 3946 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → (𝑡𝑦𝑡 ∈ On))
31 ontri1 6212 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ On ∧ 𝑠 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑠𝑡))
3231ancoms 462 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑠𝑡))
33 vex 3483 . . . . . . . . . . . . . . . . . . . . . 22 𝑡 ∈ V
34 vex 3483 . . . . . . . . . . . . . . . . . . . . . 22 𝑠 ∈ V
3533, 34brcnv 5740 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 E 𝑠𝑠 E 𝑡)
36 epel 5456 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 E 𝑡𝑠𝑡)
3735, 36bitri 278 . . . . . . . . . . . . . . . . . . . 20 (𝑡 E 𝑠𝑠𝑡)
3837notbii 323 . . . . . . . . . . . . . . . . . . 19 𝑡 E 𝑠 ↔ ¬ 𝑠𝑡)
3932, 38syl6bbr 292 . . . . . . . . . . . . . . . . . 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 610 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → ((𝑠𝑦𝑡𝑦) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠)))
4241impl 459 . . . . . . . . . . . . . . 15 (((𝑦 ⊆ On ∧ 𝑠𝑦) ∧ 𝑡𝑦) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠))
4342ralbidva 3191 . . . . . . . . . . . . . 14 ((𝑦 ⊆ On ∧ 𝑠𝑦) → (∀𝑡𝑦 𝑡𝑠 ↔ ∀𝑡𝑦 ¬ 𝑡 E 𝑠))
4443rexbidva 3288 . . . . . . . . . . . . 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 327 . . . . . . . . . . 11 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
47 vex 3483 . . . . . . . . . . . . 13 𝑦 ∈ V
4847a1i 11 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ V)
49 epweon 7491 . . . . . . . . . . . . . . . . . 18 E We On
50 wess 5529 . . . . . . . . . . . . . . . . . 18 (𝑦 ⊆ On → ( E We On → E We 𝑦))
5149, 50mpi 20 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → E We 𝑦)
52 weso 5533 . . . . . . . . . . . . . . . . 17 ( E We 𝑦 → E Or 𝑦)
5351, 52syl 17 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → E Or 𝑦)
54 cnvso 6126 . . . . . . . . . . . . . . . 16 ( E Or 𝑦 E Or 𝑦)
5553, 54sylib 221 . . . . . . . . . . . . . . 15 (𝑦 ⊆ On → E Or 𝑦)
56 onssnum 9464 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
5747, 56mpan 689 . . . . . . . . . . . . . . . . . 18 (𝑦 ⊆ On → 𝑦 ∈ dom card)
58 cardid2 9379 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
59 ensym 8554 . . . . . . . . . . . . . . . . . 18 ((card‘𝑦) ≈ 𝑦𝑦 ≈ (card‘𝑦))
6057, 58, 593syl 18 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → 𝑦 ≈ (card‘𝑦))
61 nnsdom 9114 . . . . . . . . . . . . . . . . 17 ((card‘𝑦) ∈ ω → (card‘𝑦) ≺ ω)
62 ensdomtr 8650 . . . . . . . . . . . . . . . . 17 ((𝑦 ≈ (card‘𝑦) ∧ (card‘𝑦) ≺ ω) → 𝑦 ≺ ω)
6360, 61, 62syl2an 598 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ≺ ω)
64 isfinite 9112 . . . . . . . . . . . . . . . 16 (𝑦 ∈ Fin ↔ 𝑦 ≺ ω)
6563, 64sylibr 237 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
66 wofi 8764 . . . . . . . . . . . . . . 15 (( E Or 𝑦𝑦 ∈ Fin) → E We 𝑦)
6755, 65, 66syl2an2r 684 . . . . . . . . . . . . . 14 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → E We 𝑦)
689, 67sylan 583 . . . . . . . . . . . . 13 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → E We 𝑦)
69 wefr 5532 . . . . . . . . . . . . 13 ( E We 𝑦 E Fr 𝑦)
7068, 69syl 17 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → E Fr 𝑦)
71 ssidd 3976 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦𝑦)
72 unieq 4835 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → 𝑦 = ∅)
73 uni0 4852 . . . . . . . . . . . . . . . . . . 19 ∅ = ∅
7472, 73syl6eq 2875 . . . . . . . . . . . . . . . . . 18 (𝑦 = ∅ → 𝑦 = ∅)
75 eqeq1 2828 . . . . . . . . . . . . . . . . . 18 ( 𝑦 = 𝐴 → ( 𝑦 = ∅ ↔ 𝐴 = ∅))
7674, 75syl5ib 247 . . . . . . . . . . . . . . . . 17 ( 𝑦 = 𝐴 → (𝑦 = ∅ → 𝐴 = ∅))
77 nlim0 6236 . . . . . . . . . . . . . . . . . 18 ¬ Lim ∅
78 limeq 6190 . . . . . . . . . . . . . . . . . 18 (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
7977, 78mtbiri 330 . . . . . . . . . . . . . . . . 17 (𝐴 = ∅ → ¬ Lim 𝐴)
8076, 79syl6 35 . . . . . . . . . . . . . . . 16 ( 𝑦 = 𝐴 → (𝑦 = ∅ → ¬ Lim 𝐴))
8180necon2ad 3029 . . . . . . . . . . . . . . 15 ( 𝑦 = 𝐴 → (Lim 𝐴𝑦 ≠ ∅))
8281impcom 411 . . . . . . . . . . . . . 14 ((Lim 𝐴 𝑦 = 𝐴) → 𝑦 ≠ ∅)
83823adant2 1128 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → 𝑦 ≠ ∅)
8483adantr 484 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ≠ ∅)
85 fri 5504 . . . . . . . . . . . 12 (((𝑦 ∈ V ∧ E Fr 𝑦) ∧ (𝑦𝑦𝑦 ≠ ∅)) → ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
8648, 70, 71, 84, 85syl22anc 837 . . . . . . . . . . 11 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
8746, 86mtand 815 . . . . . . . . . 10 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ (card‘𝑦) ∈ ω)
88 cardon 9370 . . . . . . . . . . 11 (card‘𝑦) ∈ On
89 eloni 6188 . . . . . . . . . . 11 ((card‘𝑦) ∈ On → Ord (card‘𝑦))
90 ordom 7583 . . . . . . . . . . . 12 Ord ω
91 ordtri1 6211 . . . . . . . . . . . 12 ((Ord ω ∧ Ord (card‘𝑦)) → (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω))
9290, 91mpan 689 . . . . . . . . . . 11 (Ord (card‘𝑦) → (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω))
9388, 89, 92mp2b 10 . . . . . . . . . 10 (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω)
9487, 93sylibr 237 . . . . . . . . 9 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ω ⊆ (card‘𝑦))
952, 94syl3an2b 1401 . . . . . . . 8 ((Lim 𝐴𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴) → ω ⊆ (card‘𝑦))
96953expb 1117 . . . . . . 7 ((Lim 𝐴 ∧ (𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴)) → ω ⊆ (card‘𝑦))
971, 96sylan2b 596 . . . . . 6 ((Lim 𝐴𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}) → ω ⊆ (card‘𝑦))
9897ralrimiva 3177 . . . . 5 (Lim 𝐴 → ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
99 ssiin 4965 . . . . 5 (ω ⊆ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦) ↔ ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
10098, 99sylibr 237 . . . 4 (Lim 𝐴 → ω ⊆ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦))
101 cflim2.1 . . . . 5 𝐴 ∈ V
102101cflim3 9682 . . . 4 (Lim 𝐴 → (cf‘𝐴) = 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦))
103100, 102sseqtrrd 3994 . . 3 (Lim 𝐴 → ω ⊆ (cf‘𝐴))
104 fvex 6674 . . . . . . 7 (card‘𝑦) ∈ V
105104dfiin2 4945 . . . . . 6 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦) = {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)}
106102, 105syl6eq 2875 . . . . 5 (Lim 𝐴 → (cf‘𝐴) = {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
107 cardlim 9398 . . . . . . . . 9 (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))
108 sseq2 3979 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ ω ⊆ (card‘𝑦)))
109 limeq 6190 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (Lim 𝑥 ↔ Lim (card‘𝑦)))
110108, 109bibi12d 349 . . . . . . . . 9 (𝑥 = (card‘𝑦) → ((ω ⊆ 𝑥 ↔ Lim 𝑥) ↔ (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))))
111107, 110mpbiri 261 . . . . . . . 8 (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
112111rexlimivw 3274 . . . . . . 7 (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
113112ss2abi 4029 . . . . . 6 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)}
114 eleq1 2903 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
11588, 114mpbiri 261 . . . . . . . . 9 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
116115rexlimivw 3274 . . . . . . . 8 (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦) → 𝑥 ∈ On)
117116abssi 4032 . . . . . . 7 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ On
118 fvex 6674 . . . . . . . . 9 (cf‘𝐴) ∈ V
119106, 118eqeltrrdi 2925 . . . . . . . 8 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
120 intex 5226 . . . . . . . 8 ({𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅ ↔ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
121119, 120sylibr 237 . . . . . . 7 (Lim 𝐴 → {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅)
122 onint 7504 . . . . . . 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)})
123117, 121, 122sylancr 590 . . . . . 6 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
124113, 123sseldi 3951 . . . . 5 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)})
125106, 124eqeltrd 2916 . . . 4 (Lim 𝐴 → (cf‘𝐴) ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)})
126 sseq2 3979 . . . . . 6 (𝑥 = (cf‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (cf‘𝐴)))
127 limeq 6190 . . . . . 6 (𝑥 = (cf‘𝐴) → (Lim 𝑥 ↔ Lim (cf‘𝐴)))
128126, 127bibi12d 349 . . . . 5 (𝑥 = (cf‘𝐴) → ((ω ⊆ 𝑥 ↔ Lim 𝑥) ↔ (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴))))
129118, 128elab 3653 . . . 4 ((cf‘𝐴) ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)} ↔ (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴)))
130125, 129sylib 221 . . 3 (Lim 𝐴 → (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴)))
131103, 130mpbid 235 . 2 (Lim 𝐴 → Lim (cf‘𝐴))
132 eloni 6188 . . . . . . 7 (𝐴 ∈ On → Ord 𝐴)
133 ordzsl 7554 . . . . . . 7 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
134132, 133sylib 221 . . . . . 6 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
135 df-3or 1085 . . . . . . 7 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ∨ Lim 𝐴))
136 orcom 867 . . . . . . 7 (((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ∨ Lim 𝐴) ↔ (Lim 𝐴 ∨ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
137 df-or 845 . . . . . . 7 ((Lim 𝐴 ∨ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
138135, 136, 1373bitri 300 . . . . . 6 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
139134, 138sylib 221 . . . . 5 (𝐴 ∈ On → (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
140 fveq2 6661 . . . . . . . . 9 (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
141 cf0 9671 . . . . . . . . 9 (cf‘∅) = ∅
142140, 141syl6eq 2875 . . . . . . . 8 (𝐴 = ∅ → (cf‘𝐴) = ∅)
143 limeq 6190 . . . . . . . 8 ((cf‘𝐴) = ∅ → (Lim (cf‘𝐴) ↔ Lim ∅))
144142, 143syl 17 . . . . . . 7 (𝐴 = ∅ → (Lim (cf‘𝐴) ↔ Lim ∅))
14577, 144mtbiri 330 . . . . . 6 (𝐴 = ∅ → ¬ Lim (cf‘𝐴))
146 1n0 8115 . . . . . . . . . 10 1o ≠ ∅
147 df1o2 8112 . . . . . . . . . . . 12 1o = {∅}
148147unieqi 4837 . . . . . . . . . . 11 1o = {∅}
149 0ex 5197 . . . . . . . . . . . 12 ∅ ∈ V
150149unisn 4844 . . . . . . . . . . 11 {∅} = ∅
151148, 150eqtri 2847 . . . . . . . . . 10 1o = ∅
152146, 151neeqtrri 3087 . . . . . . . . 9 1o 1o
153 limuni 6238 . . . . . . . . . 10 (Lim 1o → 1o = 1o)
154153necon3ai 3039 . . . . . . . . 9 (1o 1o → ¬ Lim 1o)
155152, 154ax-mp 5 . . . . . . . 8 ¬ Lim 1o
156 fveq2 6661 . . . . . . . . . 10 (𝐴 = suc 𝑥 → (cf‘𝐴) = (cf‘suc 𝑥))
157 cfsuc 9677 . . . . . . . . . 10 (𝑥 ∈ On → (cf‘suc 𝑥) = 1o)
158156, 157sylan9eqr 2881 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (cf‘𝐴) = 1o)
159 limeq 6190 . . . . . . . . 9 ((cf‘𝐴) = 1o → (Lim (cf‘𝐴) ↔ Lim 1o))
160158, 159syl 17 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (Lim (cf‘𝐴) ↔ Lim 1o))
161155, 160mtbiri 330 . . . . . . 7 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → ¬ Lim (cf‘𝐴))
162161rexlimiva 3273 . . . . . 6 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ Lim (cf‘𝐴))
163145, 162jaoi 854 . . . . 5 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → ¬ Lim (cf‘𝐴))
164139, 163syl6 35 . . . 4 (𝐴 ∈ On → (¬ Lim 𝐴 → ¬ Lim (cf‘𝐴)))
165164con4d 115 . . 3 (𝐴 ∈ On → (Lim (cf‘𝐴) → Lim 𝐴))
166 cff 9668 . . . . . . . . 9 cf:On⟶On
167166fdmi 6514 . . . . . . . 8 dom cf = On
168167eleq2i 2907 . . . . . . 7 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
169 ndmfv 6691 . . . . . . 7 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
170168, 169sylnbir 334 . . . . . 6 𝐴 ∈ On → (cf‘𝐴) = ∅)
171170, 143syl 17 . . . . 5 𝐴 ∈ On → (Lim (cf‘𝐴) ↔ Lim ∅))
17277, 171mtbiri 330 . . . 4 𝐴 ∈ On → ¬ Lim (cf‘𝐴))
173172pm2.21d 121 . . 3 𝐴 ∈ On → (Lim (cf‘𝐴) → Lim 𝐴))
174165, 173pm2.61i 185 . 2 (Lim (cf‘𝐴) → Lim 𝐴)
175131, 174impbii 212 1 (Lim 𝐴 ↔ Lim (cf‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3o 1083  w3a 1084   = wceq 1538  wcel 2115  {cab 2802  wne 3014  wral 3133  wrex 3134  {crab 3137  Vcvv 3480  wss 3919  c0 4276  𝒫 cpw 4522  {csn 4550   cuni 4824   cint 4862   ciin 4906   class class class wbr 5052   E cep 5451   Or wor 5460   Fr wfr 5498   We wwe 5500  ccnv 5541  dom cdm 5542  Ord word 6177  Oncon0 6178  Lim wlim 6179  suc csuc 6180  cfv 6343  ωcom 7574  1oc1o 8091  cen 8502  csdm 8504  Fincfn 8505  cardccrd 9361  cfccf 9363
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-card 9365  df-cf 9367
This theorem is referenced by:  cfom  9684
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