MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  scott0s Structured version   Visualization version   GIF version

Theorem scott0s 8703
Description: Theorem scheme version of scott0 8701. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is not empty iff there is an 𝑥 such that 𝜑(𝑥) holds. (Contributed by NM, 13-Oct-2003.)
Assertion
Ref Expression
scott0s (∃𝑥𝜑 ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem scott0s
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 abn0 3933 . 2 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
2 scott0 8701 . . . 4 ({𝑥𝜑} = ∅ ↔ {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
3 nfcv 2761 . . . . . . 7 𝑧{𝑥𝜑}
4 nfab1 2763 . . . . . . 7 𝑥{𝑥𝜑}
5 nfv 1840 . . . . . . . 8 𝑥(rank‘𝑧) ⊆ (rank‘𝑦)
64, 5nfral 2940 . . . . . . 7 𝑥𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)
7 nfv 1840 . . . . . . 7 𝑧𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)
8 fveq2 6153 . . . . . . . . 9 (𝑧 = 𝑥 → (rank‘𝑧) = (rank‘𝑥))
98sseq1d 3616 . . . . . . . 8 (𝑧 = 𝑥 → ((rank‘𝑧) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
109ralbidv 2981 . . . . . . 7 (𝑧 = 𝑥 → (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)))
113, 4, 6, 7, 10cbvrab 3187 . . . . . 6 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)}
12 df-rab 2916 . . . . . 6 {𝑥 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))}
13 abid 2609 . . . . . . . 8 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
14 df-ral 2912 . . . . . . . . 9 (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
15 df-sbc 3422 . . . . . . . . . . 11 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
1615imbi1i 339 . . . . . . . . . 10 (([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
1716albii 1744 . . . . . . . . 9 (∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
1814, 17bitr4i 267 . . . . . . . 8 (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))
1913, 18anbi12i 732 . . . . . . 7 ((𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))))
2019abbii 2736 . . . . . 6 {𝑥 ∣ (𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
2111, 12, 203eqtri 2647 . . . . 5 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
2221eqeq1i 2626 . . . 4 ({𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = ∅)
232, 22bitri 264 . . 3 ({𝑥𝜑} = ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = ∅)
2423necon3bii 2842 . 2 ({𝑥𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
251, 24bitr3i 266 1 (∃𝑥𝜑 ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  {crab 2911  [wsbc 3421  wss 3559  c0 3896  cfv 5852  rankcrnk 8578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-r1 8579  df-rank 8580
This theorem is referenced by:  hta  8712
  Copyright terms: Public domain W3C validator