Theorem nnwosdc 12179

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 3475	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∃𝑥 ∈ ℕ 𝜑)
2		ssrab2 3265	. . . . . 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 1955	. . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑗 / 𝑥]𝜑
9	8	nfdc 1670	. . . . . . . . 9 ⊢ Ⅎ𝑥DECID [𝑗 / 𝑥]𝜑
10		sbequ12 1782	. . . . . . . . . 10 ⊢ (𝑥 = 𝑗 → (𝜑 ↔ [𝑗 / 𝑥]𝜑))
11	10	dcbid 839	. . . . . . . . 9 ⊢ (𝑥 = 𝑗 → (DECID 𝜑 ↔ DECID [𝑗 / 𝑥]𝜑))
12	9, 11	rspc 2859	. . . . . . . 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 2336	. . . . . . . 8 ⊢ Ⅎ𝑥𝑗
16		nfcv 2336	. . . . . . . 8 ⊢ Ⅎ𝑥ℕ
17	15, 16, 8, 10	elrabf 2915	. . . . . . 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 2567	. . . 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 2674	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ ℕ ∣ 𝜑}
25		nfcv 2336	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑}
26	24, 25	nnwofdc 12178	. . 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 2478	. . . 4 ⊢ (∃𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦))
28		rabid 2670	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑥 ∈ ℕ ∧ 𝜑))
29		df-ral 2477	. . . . . . 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 2917	. . . . . . . . . 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 1481	. . . . . . 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 1616	. . . 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 2477	. . . . . . . 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 1616	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) ↔ ∃𝑥(𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))))
45		df-rex 2478	. . . . 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 1503  [wsb 1773   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  {crab 2476   ⊆ wss 3154   class class class wbr 4030   ≤ cle 8057  ℕcn 8984

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078  df-fzo 10212

This theorem is referenced by:  infpnlem2  12501