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Theorem cfeq0 9667
Description: Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
Assertion
Ref Expression
cfeq0 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))

Proof of Theorem cfeq0
Dummy variables 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 9658 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21eqeq1d 2823 . . 3 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅))
3 vex 3498 . . . . . . . . 9 𝑣 ∈ V
4 eqeq1 2825 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
54anbi1d 629 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
65exbidv 1913 . . . . . . . . 9 (𝑥 = 𝑣 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
73, 6elab 3666 . . . . . . . 8 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
8 fveq2 6664 . . . . . . . . . . . 12 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
9 cardidm 9377 . . . . . . . . . . . 12 (card‘(card‘𝑦)) = (card‘𝑦)
108, 9syl6eq 2872 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
11 eqeq2 2833 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
1210, 11mpbird 258 . . . . . . . . . 10 (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
1312adantr 481 . . . . . . . . 9 ((𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
1413exlimiv 1922 . . . . . . . 8 (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
157, 14sylbi 218 . . . . . . 7 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → (card‘𝑣) = 𝑣)
16 cardon 9362 . . . . . . 7 (card‘𝑣) ∈ On
1715, 16syl6eqelr 2922 . . . . . 6 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝑣 ∈ On)
1817ssriv 3970 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
19 onint0 7499 . . . . 5 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On → ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))}))
2018, 19ax-mp 5 . . . 4 ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 0ex 5203 . . . . . 6 ∅ ∈ V
22 eqeq1 2825 . . . . . . . 8 (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
2322anbi1d 629 . . . . . . 7 (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2423exbidv 1913 . . . . . 6 (𝑥 = ∅ → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2521, 24elab 3666 . . . . 5 (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
26 onss 7493 . . . . . . . . . . 11 (𝐴 ∈ On → 𝐴 ⊆ On)
27 sstr 3974 . . . . . . . . . . . 12 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
2827ancoms 459 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
2926, 28sylan 580 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
30293adant2 1123 . . . . . . . . 9 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ 𝑦𝐴) → 𝑦 ⊆ On)
31303adant3r 1173 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
32 simp2 1129 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → ∅ = (card‘𝑦))
33 simp3 1130 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
34 eqcom 2828 . . . . . . . . . . . 12 (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
35 vex 3498 . . . . . . . . . . . . . 14 𝑦 ∈ V
36 onssnum 9455 . . . . . . . . . . . . . 14 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
3735, 36mpan 686 . . . . . . . . . . . . 13 (𝑦 ⊆ On → 𝑦 ∈ dom card)
38 cardnueq0 9382 . . . . . . . . . . . . 13 (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
3937, 38syl 17 . . . . . . . . . . . 12 (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4034, 39syl5bb 284 . . . . . . . . . . 11 (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
4140biimpa 477 . . . . . . . . . 10 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
42 sseq1 3991 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦𝐴 ↔ ∅ ⊆ 𝐴))
43 rexeq 3407 . . . . . . . . . . . . 13 (𝑦 = ∅ → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧𝑤))
4443ralbidv 3197 . . . . . . . . . . . 12 (𝑦 = ∅ → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
4542, 44anbi12d 630 . . . . . . . . . . 11 (𝑦 = ∅ → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)))
4645biimpa 477 . . . . . . . . . 10 ((𝑦 = ∅ ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
4741, 46sylan 580 . . . . . . . . 9 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
48 rex0 4316 . . . . . . . . . . . . 13 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
4948rgenw 3150 . . . . . . . . . . . 12 𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
50 r19.2z 4438 . . . . . . . . . . . 12 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤) → ∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
5149, 50mpan2 687 . . . . . . . . . . 11 (𝐴 ≠ ∅ → ∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
52 rexnal 3238 . . . . . . . . . . 11 (∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤 ↔ ¬ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)
5351, 52sylib 219 . . . . . . . . . 10 (𝐴 ≠ ∅ → ¬ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)
5453necon4ai 3047 . . . . . . . . 9 (∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤𝐴 = ∅)
5547, 54simpl2im 504 . . . . . . . 8 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅)
5631, 32, 33, 55syl21anc 833 . . . . . . 7 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅)
57563expib 1114 . . . . . 6 (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅))
5857exlimdv 1925 . . . . 5 (𝐴 ∈ On → (∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅))
5925, 58syl5bi 243 . . . 4 (𝐴 ∈ On → (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝐴 = ∅))
6020, 59syl5bi 243 . . 3 (𝐴 ∈ On → ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ → 𝐴 = ∅))
612, 60sylbid 241 . 2 (𝐴 ∈ On → ((cf‘𝐴) = ∅ → 𝐴 = ∅))
62 fveq2 6664 . . 3 (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
63 cf0 9662 . . 3 (cf‘∅) = ∅
6462, 63syl6eq 2872 . 2 (𝐴 = ∅ → (cf‘𝐴) = ∅)
6561, 64impbid1 226 1 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  {cab 2799  wne 3016  wral 3138  wrex 3139  Vcvv 3495  wss 3935  c0 4290   cint 4869  dom cdm 5549  Oncon0 6185  cfv 6349  cardccrd 9353  cfccf 9355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-wrecs 7938  df-recs 7999  df-er 8279  df-en 8499  df-dom 8500  df-card 9357  df-cf 9359
This theorem is referenced by: (None)
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