Theorem nnwosdc 12206

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 3478	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∃𝑥 ∈ ℕ 𝜑)
2		ssrab2 3268	. . . . . 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 827	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℕ DECID 𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 ∈ ℕ ∨ ¬ 𝑗 ∈ ℕ))
6		df-dc 836	. . . . . . . 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 1958	. . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑗 / 𝑥]𝜑
9	8	nfdc 1673	. . . . . . . . 9 ⊢ Ⅎ𝑥DECID [𝑗 / 𝑥]𝜑
10		sbequ12 1785	. . . . . . . . . 10 ⊢ (𝑥 = 𝑗 → (𝜑 ↔ [𝑗 / 𝑥]𝜑))
11	10	dcbid 839	. . . . . . . . 9 ⊢ (𝑥 = 𝑗 → (DECID 𝜑 ↔ DECID [𝑗 / 𝑥]𝜑))
12	9, 11	rspc 2862	. . . . . . . 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 934	. . . . . 6 ⊢ ((∀𝑥 ∈ ℕ DECID 𝜑 ∧ 𝑗 ∈ ℕ) → DECID (𝑗 ∈ ℕ ∧ [𝑗 / 𝑥]𝜑))
15		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝑗
16		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥ℕ
17	15, 16, 8, 10	elrabf 2918	. . . . . . 7 ⊢ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑗 ∈ ℕ ∧ [𝑗 / 𝑥]𝜑))
18	17	dcbii 841	. . . . . 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 2570	. . . 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 982	. . 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 2677	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ ℕ ∣ 𝜑}
25		nfcv 2339	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑}
26	24, 25	nnwofdc 12205	. . 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 2481	. . . 4 ⊢ (∃𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦))
28		rabid 2673	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑥 ∈ ℕ ∧ 𝜑))
29		df-ral 2480	. . . . . . 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 2920	. . . . . . . . . 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 1484	. . . . . . 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 1619	. . . 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 2480	. . . . . . . 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 1619	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) ↔ ∃𝑥(𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))))
45		df-rex 2481	. . . . 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 709  DECID wdc 835   ∧ w3a 980  ∀wal 1362  ∃wex 1506  [wsb 1776   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479   ⊆ wss 3157   class class class wbr 4033   ≤ cle 8062  ℕcn 8990

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218

This theorem is referenced by:  infpnlem2  12529