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Theorem scott0 8746
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 3190 . . 3 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
2 rab0 3953 . . 3 {𝑥 ∈ ∅ ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅
31, 2syl6eq 2671 . 2 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
4 n0 3929 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 nfre1 3004 . . . . . . . . 9 𝑥𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)
6 eqid 2621 . . . . . . . . . 10 (rank‘𝑥) = (rank‘𝑥)
7 rspe 3002 . . . . . . . . . 10 ((𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
86, 7mpan2 707 . . . . . . . . 9 (𝑥𝐴 → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
95, 8exlimi 2085 . . . . . . . 8 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
104, 9sylbi 207 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
11 fvex 6199 . . . . . . . . . . 11 (rank‘𝑥) ∈ V
12 eqeq1 2625 . . . . . . . . . . . 12 (𝑦 = (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ (rank‘𝑥) = (rank‘𝑥)))
1312anbi2d 740 . . . . . . . . . . 11 (𝑦 = (rank‘𝑥) → ((𝑥𝐴𝑦 = (rank‘𝑥)) ↔ (𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥))))
1411, 13spcev 3298 . . . . . . . . . 10 ((𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
1514eximi 1761 . . . . . . . . 9 (∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
16 excom 2041 . . . . . . . . 9 (∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)) ↔ ∃𝑥𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
1715, 16sylibr 224 . . . . . . . 8 (∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
18 df-rex 2917 . . . . . . . 8 (∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) ↔ ∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)))
19 df-rex 2917 . . . . . . . . 9 (∃𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
2019exbii 1773 . . . . . . . 8 (∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
2117, 18, 203imtr4i 281 . . . . . . 7 (∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
2210, 21syl 17 . . . . . 6 (𝐴 ≠ ∅ → ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
23 abn0 3952 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ ↔ ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
2422, 23sylibr 224 . . . . 5 (𝐴 ≠ ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅)
2511dfiin2 4553 . . . . . 6 𝑥𝐴 (rank‘𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)}
26 rankon 8655 . . . . . . . . . 10 (rank‘𝑥) ∈ On
27 eleq1 2688 . . . . . . . . . 10 (𝑦 = (rank‘𝑥) → (𝑦 ∈ On ↔ (rank‘𝑥) ∈ On))
2826, 27mpbiri 248 . . . . . . . . 9 (𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
2928rexlimivw 3027 . . . . . . . 8 (∃𝑥𝐴 𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
3029abssi 3675 . . . . . . 7 {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ⊆ On
31 onint 6992 . . . . . . 7 (({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
3230, 31mpan 706 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
3325, 32syl5eqel 2704 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
34 nfii1 4549 . . . . . . . . 9 𝑥 𝑥𝐴 (rank‘𝑥)
3534nfeq2 2779 . . . . . . . 8 𝑥 𝑦 = 𝑥𝐴 (rank‘𝑥)
36 eqeq1 2625 . . . . . . . 8 (𝑦 = 𝑥𝐴 (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3735, 36rexbid 3049 . . . . . . 7 (𝑦 = 𝑥𝐴 (rank‘𝑥) → (∃𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3837elabg 3349 . . . . . 6 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} → ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ↔ ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3938ibi 256 . . . . 5 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} → ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
40 ssid 3622 . . . . . . . . . 10 (rank‘𝑦) ⊆ (rank‘𝑦)
41 fveq2 6189 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
4241sseq1d 3630 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑦) ⊆ (rank‘𝑦)))
4342rspcev 3307 . . . . . . . . . 10 ((𝑦𝐴 ∧ (rank‘𝑦) ⊆ (rank‘𝑦)) → ∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
4440, 43mpan2 707 . . . . . . . . 9 (𝑦𝐴 → ∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
45 iinss 4569 . . . . . . . . 9 (∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
4644, 45syl 17 . . . . . . . 8 (𝑦𝐴 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
47 sseq1 3624 . . . . . . . 8 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ( 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
4846, 47syl5ib 234 . . . . . . 7 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → (𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
4948ralrimiv 2964 . . . . . 6 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5049reximi 3010 . . . . 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 3956 . . . 4 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑥𝐴𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5351, 52sylibr 224 . . 3 (𝐴 ≠ ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
5453necon4i 2828 . 2 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ → 𝐴 = ∅)
553, 54impbii 199 1 (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1482  wex 1703  wcel 1989  {cab 2607  wne 2793  wral 2911  wrex 2912  {crab 2915  wss 3572  c0 3913   cint 4473   ciin 4519  Oncon0 5721  cfv 5886  rankcrnk 8623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-r1 8624  df-rank 8625
This theorem is referenced by:  scott0s  8748  cplem1  8749  karden  8755  scott0f  33957
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