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Theorem bj-nn0suc 7329
Description: Proof of (biconditional form of) nn0suc 4254 from the core axioms of CZF. See also bj-nn0sucALT 7343. 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 7319 . . 3 (A 𝜔 → (A = ∅ x A A = suc x))
2 bj-omtrans 7325 . . . . 5 (A 𝜔 → A ⊆ 𝜔)
3 ssrexv 2982 . . . . 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 689 . . 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 4244 . . . 4 𝜔
8 eleq1 2082 . . . 4 (A = ∅ → (A 𝜔 ↔ ∅ 𝜔))
97, 8mpbiri 157 . . 3 (A = ∅ → A 𝜔)
10 bj-peano2 7308 . . . . 5 (x 𝜔 → suc x 𝜔)
11 eleq1a 2091 . . . . . 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 2406 . . 3 (x 𝜔 A = suc xA 𝜔)
159, 14jaoi 623 . 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 616   = wceq 1228   wcel 1374  wrex 2285  wss 2894  c0 3201  suc csuc 4051  𝜔com 4240
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7187  ax-bdim 7188  ax-bdan 7189  ax-bdor 7190  ax-bdn 7191  ax-bdal 7192  ax-bdex 7193  ax-bdeq 7194  ax-bdel 7195  ax-bdsb 7196  ax-bdsep 7258  ax-infvn 7310
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7215  df-bj-ind 7297
This theorem is referenced by:  bj-findis  7344
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