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Theorem scott0 8994
Description: Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. 𝐴 is empty). (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
scott0 (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem scott0
StepHypRef Expression
1 rabeq 3380 . . 3 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
2 rab0 4154 . . 3 {𝑥 ∈ ∅ ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅
31, 2syl6eq 2854 . 2 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
4 n0 4130 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 nfre1 3190 . . . . . . . . 9 𝑥𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)
6 eqid 2804 . . . . . . . . . 10 (rank‘𝑥) = (rank‘𝑥)
7 rspe 3188 . . . . . . . . . 10 ((𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
86, 7mpan2 674 . . . . . . . . 9 (𝑥𝐴 → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
95, 8exlimi 2253 . . . . . . . 8 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
104, 9sylbi 208 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
11 fvex 6419 . . . . . . . . . . 11 (rank‘𝑥) ∈ V
12 eqeq1 2808 . . . . . . . . . . . 12 (𝑦 = (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ (rank‘𝑥) = (rank‘𝑥)))
1312anbi2d 616 . . . . . . . . . . 11 (𝑦 = (rank‘𝑥) → ((𝑥𝐴𝑦 = (rank‘𝑥)) ↔ (𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥))))
1411, 13spcev 3491 . . . . . . . . . 10 ((𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
1514eximi 1919 . . . . . . . . 9 (∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
16 excom 2209 . . . . . . . . 9 (∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)) ↔ ∃𝑥𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
1715, 16sylibr 225 . . . . . . . 8 (∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
18 df-rex 3100 . . . . . . . 8 (∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) ↔ ∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)))
19 df-rex 3100 . . . . . . . . 9 (∃𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
2019exbii 1933 . . . . . . . 8 (∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
2117, 18, 203imtr4i 283 . . . . . . 7 (∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
2210, 21syl 17 . . . . . 6 (𝐴 ≠ ∅ → ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
23 abn0 4153 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ ↔ ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
2422, 23sylibr 225 . . . . 5 (𝐴 ≠ ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅)
2511dfiin2 4745 . . . . . 6 𝑥𝐴 (rank‘𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)}
26 rankon 8903 . . . . . . . . . 10 (rank‘𝑥) ∈ On
27 eleq1 2871 . . . . . . . . . 10 (𝑦 = (rank‘𝑥) → (𝑦 ∈ On ↔ (rank‘𝑥) ∈ On))
2826, 27mpbiri 249 . . . . . . . . 9 (𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
2928rexlimivw 3215 . . . . . . . 8 (∃𝑥𝐴 𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
3029abssi 3872 . . . . . . 7 {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ⊆ On
31 onint 7223 . . . . . . 7 (({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
3230, 31mpan 673 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
3325, 32syl5eqel 2887 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
34 nfii1 4741 . . . . . . . . 9 𝑥 𝑥𝐴 (rank‘𝑥)
3534nfeq2 2962 . . . . . . . 8 𝑥 𝑦 = 𝑥𝐴 (rank‘𝑥)
36 eqeq1 2808 . . . . . . . 8 (𝑦 = 𝑥𝐴 (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3735, 36rexbid 3237 . . . . . . 7 (𝑦 = 𝑥𝐴 (rank‘𝑥) → (∃𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3837elabg 3544 . . . . . 6 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} → ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ↔ ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3938ibi 258 . . . . 5 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} → ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
40 ssid 3818 . . . . . . . . . 10 (rank‘𝑦) ⊆ (rank‘𝑦)
41 fveq2 6406 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
4241sseq1d 3827 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑦) ⊆ (rank‘𝑦)))
4342rspcev 3500 . . . . . . . . . 10 ((𝑦𝐴 ∧ (rank‘𝑦) ⊆ (rank‘𝑦)) → ∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
4440, 43mpan2 674 . . . . . . . . 9 (𝑦𝐴 → ∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
45 iinss 4761 . . . . . . . . 9 (∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
4644, 45syl 17 . . . . . . . 8 (𝑦𝐴 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
47 sseq1 3821 . . . . . . . 8 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ( 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
4846, 47syl5ib 235 . . . . . . 7 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → (𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
4948ralrimiv 3151 . . . . . 6 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5049reximi 3196 . . . . 5 (∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∃𝑥𝐴𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5124, 33, 39, 504syl 19 . . . 4 (𝐴 ≠ ∅ → ∃𝑥𝐴𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
52 rabn0 4156 . . . 4 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑥𝐴𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5351, 52sylibr 225 . . 3 (𝐴 ≠ ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
5453necon4i 3011 . 2 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ → 𝐴 = ∅)
553, 54impbii 200 1 (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wex 1859  wcel 2156  {cab 2790  wne 2976  wral 3094  wrex 3095  {crab 3098  wss 3767  c0 4114   cint 4667   ciin 4711  Oncon0 5934  cfv 6099  rankcrnk 8871
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-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177
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-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-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-om 7294  df-wrecs 7640  df-recs 7702  df-rdg 7740  df-r1 8872  df-rank 8873
This theorem is referenced by:  scott0s  8996  cplem1  8997  karden  9003  scott0f  34285
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