Theorem scott0s 9863

Description: Theorem scheme version of scott0 9861. 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.

Step	Hyp	Ref	Expression
1		abn0 4342	. 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
2		scott0 9861	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ {𝑧 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
3		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑧{𝑥 ∣ 𝜑}
4		nfab1 2927	. . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
5		nfv 1944	. . . . . . . 8 ⊢ Ⅎ𝑥(rank‘𝑧) ⊆ (rank‘𝑦)
6	4, 5	nfralw 3312	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)
7		nfv 1944	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)
8		fveq2 6883	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (rank‘𝑧) = (rank‘𝑥))
9	8	sseq1d 3969	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((rank‘𝑧) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
10	9	ralbidv 3188	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)))
11	3, 4, 6, 7, 10	cbvrabw 3451	. . . . . 6 ⊢ {𝑧 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)}
12		df-rab 3417	. . . . . 6 ⊢ {𝑥 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))}
13		abid 2745	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
14		df-ral 3080	. . . . . . . . 9 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
15		df-sbc 3746	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑})
16	15	imbi1i 352	. . . . . . . . . 10 ⊢ (([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
17	16	albii 1849	. . . . . . . . 9 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
18	14, 17	bitr4i 281	. . . . . . . 8 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))
19	13, 18	anbi12i 639	. . . . . . 7 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))))
20	19	abbii 2830	. . . . . 6 ⊢ {𝑥 ∣ (𝑥 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
21	11, 12, 20	3eqtri 2790	. . . . 5 ⊢ {𝑧 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
22	21	eqeq1i 2768	. . . 4 ⊢ ({𝑧 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = ∅)
23	2, 22	bitri 278	. . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = ∅)
24	23	necon3bii 3010	. 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
25	1, 24	bitr3i 280	1 ⊢ (∃𝑥𝜑 ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741   ≠ wne 2958  ∀wral 3079  {crab 3416  [wsbc 3745   ⊆ wss 3906  ∅c0 4287  ‘cfv 6538  rankcrnk 9736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-r1 9737  df-rank 9738

This theorem is referenced by:  hta  9884