Theorem nnwosdc 12615

Description: Well-ordering principle: any inhabited decidable set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 25-Oct-2024.)

Hypothesis

Ref	Expression
nnwos.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
nnwosdc	⊢ ((∃𝑥 ∈ ℕ 𝜑 ∧ ∀𝑥 ∈ ℕ DECID 𝜑) → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem nnwosdc

Dummy variables 𝑗 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabn0m 3522	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∃𝑥 ∈ ℕ 𝜑)
2		ssrab2 3312	. . . . . 6 ⊢ {𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ
3	2	biantrur 303	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
4	1, 3	sylbb1 137	. . . 4 ⊢ (∃𝑥 ∈ ℕ 𝜑 → ({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
5		animorrl 833	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℕ DECID 𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 ∈ ℕ ∨ ¬ 𝑗 ∈ ℕ))
6		df-dc 842	. . . . . . . 8 ⊢ (DECID 𝑗 ∈ ℕ ↔ (𝑗 ∈ ℕ ∨ ¬ 𝑗 ∈ ℕ))
7	5, 6	sylibr 134	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℕ DECID 𝜑 ∧ 𝑗 ∈ ℕ) → DECID 𝑗 ∈ ℕ)
8		nfs1v 1992	. . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑗 / 𝑥]𝜑
9	8	nfdc 1707	. . . . . . . . 9 ⊢ Ⅎ𝑥DECID [𝑗 / 𝑥]𝜑
10		sbequ12 1819	. . . . . . . . . 10 ⊢ (𝑥 = 𝑗 → (𝜑 ↔ [𝑗 / 𝑥]𝜑))
11	10	dcbid 845	. . . . . . . . 9 ⊢ (𝑥 = 𝑗 → (DECID 𝜑 ↔ DECID [𝑗 / 𝑥]𝜑))
12	9, 11	rspc 2904	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (∀𝑥 ∈ ℕ DECID 𝜑 → DECID [𝑗 / 𝑥]𝜑))
13	12	impcom 125	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℕ DECID 𝜑 ∧ 𝑗 ∈ ℕ) → DECID [𝑗 / 𝑥]𝜑)
14	7, 13	dcand 940	. . . . . 6 ⊢ ((∀𝑥 ∈ ℕ DECID 𝜑 ∧ 𝑗 ∈ ℕ) → DECID (𝑗 ∈ ℕ ∧ [𝑗 / 𝑥]𝜑))
15		nfcv 2374	. . . . . . . 8 ⊢ Ⅎ𝑥𝑗
16		nfcv 2374	. . . . . . . 8 ⊢ Ⅎ𝑥ℕ
17	15, 16, 8, 10	elrabf 2960	. . . . . . 7 ⊢ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑗 ∈ ℕ ∧ [𝑗 / 𝑥]𝜑))
18	17	dcbii 847	. . . . . 6 ⊢ (DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ DECID (𝑗 ∈ ℕ ∧ [𝑗 / 𝑥]𝜑))
19	14, 18	sylibr 134	. . . . 5 ⊢ ((∀𝑥 ∈ ℕ DECID 𝜑 ∧ 𝑗 ∈ ℕ) → DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑})
20	19	ralrimiva 2605	. . . 4 ⊢ (∀𝑥 ∈ ℕ DECID 𝜑 → ∀𝑗 ∈ ℕ DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑})
21	4, 20	anim12i 338	. . 3 ⊢ ((∃𝑥 ∈ ℕ 𝜑 ∧ ∀𝑥 ∈ ℕ DECID 𝜑) → (({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑}) ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
22		df-3an 1006	. . 3 ⊢ (({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑}) ↔ (({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑}) ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
23	21, 22	sylibr 134	. 2 ⊢ ((∃𝑥 ∈ ℕ 𝜑 ∧ ∀𝑥 ∈ ℕ DECID 𝜑) → ({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
24		nfrab1 2713	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ ℕ ∣ 𝜑}
25		nfcv 2374	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑}
26	24, 25	nnwofdc 12614	. . 3 ⊢ (({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ∃𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦)
27		df-rex 2516	. . . 4 ⊢ (∃𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦))
28		rabid 2709	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑥 ∈ ℕ ∧ 𝜑))
29		df-ral 2515	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∀𝑦(𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑥 ≤ 𝑦))
30		nnwos.1	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
31	30, 30, 30	3bitr2d 216	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
32	31	elrab 2962	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜓))
33	32	imbi1i 238	. . . . . . . . 9 ⊢ ((𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑥 ≤ 𝑦) ↔ ((𝑦 ∈ ℕ ∧ 𝜓) → 𝑥 ≤ 𝑦))
34		impexp 263	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℕ ∧ 𝜓) → 𝑥 ≤ 𝑦) ↔ (𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦)))
35	33, 34	bitri 184	. . . . . . . 8 ⊢ ((𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑥 ≤ 𝑦) ↔ (𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦)))
36	35	albii 1518	. . . . . . 7 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑥 ≤ 𝑦) ↔ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦)))
37	29, 36	bitri 184	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦)))
38	28, 37	anbi12i 460	. . . . 5 ⊢ ((𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦) ↔ ((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))))
39	38	exbii 1653	. . . 4 ⊢ (∃𝑥(𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦) ↔ ∃𝑥((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))))
40		df-ral 2515	. . . . . . . 8 ⊢ (∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦) ↔ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦)))
41	40	anbi2i 457	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)) ↔ ((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))))
42		anass 401	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)) ↔ (𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))))
43	41, 42	bitr3i 186	. . . . . 6 ⊢ (((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) ↔ (𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))))
44	43	exbii 1653	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) ↔ ∃𝑥(𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))))
45		df-rex 2516	. . . . 5 ⊢ (∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)) ↔ ∃𝑥(𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))))
46	44, 45	bitr4i 187	. . . 4 ⊢ (∃𝑥((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) ↔ ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))
47	27, 39, 46	3bitri 206	. . 3 ⊢ (∃𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))
48	26, 47	sylib 122	. 2 ⊢ (({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))
49	23, 48	syl 14	1 ⊢ ((∃𝑥 ∈ ℕ 𝜑 ∧ ∀𝑥 ∈ ℕ DECID 𝜑) → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  DECID wdc 841   ∧ w3a 1004  ∀wal 1395  ∃wex 1540  [wsb 1810   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  {crab 2514   ⊆ wss 3200   class class class wbr 4088   ≤ cle 8215  ℕcn 9143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378

This theorem is referenced by:  infpnlem2  12938