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Theorem bj-nn0suc 9394
Description: Proof of (biconditional form of) nn0suc 4270 from the core axioms of CZF. See also bj-nn0sucALT 9408. As a characterization of the elements of 𝜔, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nn0suc (A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x))
Distinct variable group:   x,A

Proof of Theorem bj-nn0suc
StepHypRef Expression
1 bj-nn0suc0 9384 . . 3 (A 𝜔 → (A = ∅ x A A = suc x))
2 bj-omtrans 9390 . . . . 5 (A 𝜔 → A ⊆ 𝜔)
3 ssrexv 2999 . . . . 5 (A ⊆ 𝜔 → (x A A = suc xx 𝜔 A = suc x))
42, 3syl 14 . . . 4 (A 𝜔 → (x A A = suc xx 𝜔 A = suc x))
54orim2d 701 . . 3 (A 𝜔 → ((A = ∅ x A A = suc x) → (A = ∅ x 𝜔 A = suc x)))
61, 5mpd 13 . 2 (A 𝜔 → (A = ∅ x 𝜔 A = suc x))
7 peano1 4260 . . . 4 𝜔
8 eleq1 2097 . . . 4 (A = ∅ → (A 𝜔 ↔ ∅ 𝜔))
97, 8mpbiri 157 . . 3 (A = ∅ → A 𝜔)
10 bj-peano2 9373 . . . . 5 (x 𝜔 → suc x 𝜔)
11 eleq1a 2106 . . . . . 6 (suc x 𝜔 → (A = suc xA 𝜔))
1211imp 115 . . . . 5 ((suc x 𝜔 A = suc x) → A 𝜔)
1310, 12sylan 267 . . . 4 ((x 𝜔 A = suc x) → A 𝜔)
1413rexlimiva 2422 . . 3 (x 𝜔 A = suc xA 𝜔)
159, 14jaoi 635 . 2 ((A = ∅ x 𝜔 A = suc x) → A 𝜔)
166, 15impbii 117 1 (A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wo 628   = wceq 1242   wcel 1390  wrex 2301  wss 2911  c0 3218  suc csuc 4068  𝜔com 4256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9248  ax-bdim 9249  ax-bdan 9250  ax-bdor 9251  ax-bdn 9252  ax-bdal 9253  ax-bdex 9254  ax-bdeq 9255  ax-bdel 9256  ax-bdsb 9257  ax-bdsep 9319  ax-infvn 9375
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9276  df-bj-ind 9362
This theorem is referenced by:  bj-findis  9409
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