Theorem nnwofdc 12554

Description: Well-ordering principle: any inhabited decidable set of positive integers has a least element. This version allows 𝑥 and 𝑦 to be present in 𝐴 as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nnwof.1	⊢ Ⅎ𝑥𝐴
nnwof.2	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
nnwofdc	⊢ ((𝐴 ⊆ ℕ ∧ ∃𝑧 𝑧 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)

Distinct variable groups:   𝐴,𝑗,𝑧   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nnwofdc

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

Step	Hyp	Ref	Expression
1		nnwodc 12552	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∃𝑧 𝑧 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) → ∃𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑤 ≤ 𝑣)
2		nfcv 2372	. . 3 ⊢ Ⅎ𝑤𝐴
3		nnwof.1	. . 3 ⊢ Ⅎ𝑥𝐴
4		nfv 1574	. . . 4 ⊢ Ⅎ𝑥 𝑤 ≤ 𝑣
5	3, 4	nfralw 2567	. . 3 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 𝑤 ≤ 𝑣
6		nfv 1574	. . 3 ⊢ Ⅎ𝑤∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦
7		breq1 4085	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ≤ 𝑣 ↔ 𝑥 ≤ 𝑣))
8	7	ralbidv 2530	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑣 ∈ 𝐴 𝑤 ≤ 𝑣 ↔ ∀𝑣 ∈ 𝐴 𝑥 ≤ 𝑣))
9		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑣𝐴
10		nnwof.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
11		nfv 1574	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ≤ 𝑣
12		nfv 1574	. . . . 5 ⊢ Ⅎ𝑣 𝑥 ≤ 𝑦
13		breq2 4086	. . . . 5 ⊢ (𝑣 = 𝑦 → (𝑥 ≤ 𝑣 ↔ 𝑥 ≤ 𝑦))
14	9, 10, 11, 12, 13	cbvralfw 2754	. . . 4 ⊢ (∀𝑣 ∈ 𝐴 𝑥 ≤ 𝑣 ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
15	8, 14	bitrdi 196	. . 3 ⊢ (𝑤 = 𝑥 → (∀𝑣 ∈ 𝐴 𝑤 ≤ 𝑣 ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
16	2, 3, 5, 6, 15	cbvrexfw 2755	. 2 ⊢ (∃𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑤 ≤ 𝑣 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
17	1, 16	sylib 122	1 ⊢ ((𝐴 ⊆ ℕ ∧ ∃𝑧 𝑧 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)

Syntax hints:   → wi 4  DECID wdc 839   ∧ w3a 1002  ∃wex 1538   ∈ wcel 2200  Ⅎwnfc 2359  ∀wral 2508  ∃wrex 2509   ⊆ wss 3197   class class class wbr 4082   ≤ cle 8178  ℕcn 9106

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-fzo 10335

This theorem is referenced by:  nnwosdc  12555