Theorem nnwosdc 12799

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 3549	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∃𝑥 ∈ ℕ 𝜑)
2		ssrab2 3333	. . . . . 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 838	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℕ DECID 𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 ∈ ℕ ∨ ¬ 𝑗 ∈ ℕ))
6		df-dc 847	. . . . . . . 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 1999	. . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑗 / 𝑥]𝜑
9	8	nfdc 1711	. . . . . . . . 9 ⊢ Ⅎ𝑥DECID [𝑗 / 𝑥]𝜑
10		sbequ12 1824	. . . . . . . . . 10 ⊢ (𝑥 = 𝑗 → (𝜑 ↔ [𝑗 / 𝑥]𝜑))
11	10	dcbid 850	. . . . . . . . 9 ⊢ (𝑥 = 𝑗 → (DECID 𝜑 ↔ DECID [𝑗 / 𝑥]𝜑))
12	9, 11	rspc 2923	. . . . . . . 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 945	. . . . . 6 ⊢ ((∀𝑥 ∈ ℕ DECID 𝜑 ∧ 𝑗 ∈ ℕ) → DECID (𝑗 ∈ ℕ ∧ [𝑗 / 𝑥]𝜑))
15		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥𝑗
16		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥ℕ
17	15, 16, 8, 10	elrabf 2980	. . . . . . 7 ⊢ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑗 ∈ ℕ ∧ [𝑗 / 𝑥]𝜑))
18	17	dcbii 852	. . . . . 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 2623	. . . 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 1011	. . 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 2732	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ ℕ ∣ 𝜑}
25		nfcv 2392	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑}
26	24, 25	nnwofdc 12798	. . 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 2534	. . . 4 ⊢ (∃𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦))
28		rabid 2727	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑥 ∈ ℕ ∧ 𝜑))
29		df-ral 2533	. . . . . . 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 2982	. . . . . . . . . 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 1523	. . . . . . 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 464	. . . . 5 ⊢ ((𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦) ↔ ((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))))
39	38	exbii 1658	. . . 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 2533	. . . . . . . 8 ⊢ (∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦) ↔ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦)))
41	40	anbi2i 461	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)) ↔ ((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))))
42		anass 405	. . . . . . 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 1658	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) ↔ ∃𝑥(𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))))
45		df-rex 2534	. . . . 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 720  DECID wdc 846   ∧ w3a 1009  ∀wal 1400  ∃wex 1545  [wsb 1815   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532   ⊆ wss 3220   class class class wbr 4128   ≤ cle 8355  ℕcn 9287

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395  df-fzo 10533

This theorem is referenced by:  infpnlem2  13122