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Theorem scott0s 8634
 Description: Theorem scheme version of scott0 8632. 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 3908 . 2 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
2 scott0 8632 . . . 4 ({𝑥𝜑} = ∅ ↔ {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
3 nfcv 2751 . . . . . . 7 𝑧{𝑥𝜑}
4 nfab1 2753 . . . . . . 7 𝑥{𝑥𝜑}
5 nfv 1830 . . . . . . . 8 𝑥(rank‘𝑧) ⊆ (rank‘𝑦)
64, 5nfral 2929 . . . . . . 7 𝑥𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)
7 nfv 1830 . . . . . . 7 𝑧𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)
8 fveq2 6103 . . . . . . . . 9 (𝑧 = 𝑥 → (rank‘𝑧) = (rank‘𝑥))
98sseq1d 3595 . . . . . . . 8 (𝑧 = 𝑥 → ((rank‘𝑧) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
109ralbidv 2969 . . . . . . 7 (𝑧 = 𝑥 → (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)))
113, 4, 6, 7, 10cbvrab 3171 . . . . . 6 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)}
12 df-rab 2905 . . . . . 6 {𝑥 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))}
13 abid 2598 . . . . . . . 8 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
14 df-ral 2901 . . . . . . . . 9 (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
15 df-sbc 3403 . . . . . . . . . . 11 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
1615imbi1i 338 . . . . . . . . . 10 (([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
1716albii 1737 . . . . . . . . 9 (∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
1814, 17bitr4i 266 . . . . . . . 8 (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))
1913, 18anbi12i 729 . . . . . . 7 ((𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))))
2019abbii 2726 . . . . . 6 {𝑥 ∣ (𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
2111, 12, 203eqtri 2636 . . . . 5 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
2221eqeq1i 2615 . . . 4 ({𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = ∅)
232, 22bitri 263 . . 3 ({𝑥𝜑} = ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = ∅)
2423necon3bii 2834 . 2 ({𝑥𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
251, 24bitr3i 265 1 (∃𝑥𝜑 ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  {crab 2900  [wsbc 3402   ⊆ wss 3540  ∅c0 3874  ‘cfv 5804  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511 This theorem is referenced by:  hta  8643
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