Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-inf2vnlem1 GIF version

Theorem bj-inf2vnlem1 10461
Description: Lemma for bj-inf2vn 10465. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inf2vnlem1 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem bj-inf2vnlem1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bi2 125 . . . . 5 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴))
2 jaob 641 . . . . . 6 (((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴) ↔ ((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)))
32biimpi 117 . . . . 5 (((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴) → ((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)))
4 simpl 106 . . . . . 6 (((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)) → (𝑥 = ∅ → 𝑥𝐴))
5 eleq1 2116 . . . . . 6 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ∈ 𝐴))
64, 5mpbidi 144 . . . . 5 (((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)) → (𝑥 = ∅ → ∅ ∈ 𝐴))
71, 3, 63syl 17 . . . 4 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (𝑥 = ∅ → ∅ ∈ 𝐴))
87alimi 1360 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴))
9 exim 1506 . . 3 (∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴) → (∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴))
10 0ex 3911 . . . . . 6 ∅ ∈ V
1110isseti 2580 . . . . 5 𝑥 𝑥 = ∅
12 pm2.27 39 . . . . 5 (∃𝑥 𝑥 = ∅ → ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∃𝑥∅ ∈ 𝐴))
1311, 12ax-mp 7 . . . 4 ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∃𝑥∅ ∈ 𝐴)
14 bj-ex 10268 . . . 4 (∃𝑥∅ ∈ 𝐴 → ∅ ∈ 𝐴)
1513, 14syl 14 . . 3 ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∅ ∈ 𝐴)
168, 9, 153syl 17 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∅ ∈ 𝐴)
173simprd 111 . . . . . 6 (((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴) → (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴))
181, 17syl 14 . . . . 5 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴))
1918alimi 1360 . . . 4 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴))
20 eqid 2056 . . . . 5 suc 𝑧 = suc 𝑧
21 suceq 4166 . . . . . . 7 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
2221eqeq2d 2067 . . . . . 6 (𝑦 = 𝑧 → (suc 𝑧 = suc 𝑦 ↔ suc 𝑧 = suc 𝑧))
2322rspcev 2673 . . . . 5 ((𝑧𝐴 ∧ suc 𝑧 = suc 𝑧) → ∃𝑦𝐴 suc 𝑧 = suc 𝑦)
2420, 23mpan2 409 . . . 4 (𝑧𝐴 → ∃𝑦𝐴 suc 𝑧 = suc 𝑦)
25 vex 2577 . . . . . 6 𝑧 ∈ V
2625bj-sucex 10409 . . . . 5 suc 𝑧 ∈ V
27 eqeq1 2062 . . . . . . 7 (𝑥 = suc 𝑧 → (𝑥 = suc 𝑦 ↔ suc 𝑧 = suc 𝑦))
2827rexbidv 2344 . . . . . 6 (𝑥 = suc 𝑧 → (∃𝑦𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦𝐴 suc 𝑧 = suc 𝑦))
29 eleq1 2116 . . . . . 6 (𝑥 = suc 𝑧 → (𝑥𝐴 ↔ suc 𝑧𝐴))
3028, 29imbi12d 227 . . . . 5 (𝑥 = suc 𝑧 → ((∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴) ↔ (∃𝑦𝐴 suc 𝑧 = suc 𝑦 → suc 𝑧𝐴)))
3126, 30spcv 2663 . . . 4 (∀𝑥(∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴) → (∃𝑦𝐴 suc 𝑧 = suc 𝑦 → suc 𝑧𝐴))
3219, 24, 31syl2im 38 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (𝑧𝐴 → suc 𝑧𝐴))
3332ralrimiv 2408 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑧𝐴 suc 𝑧𝐴)
34 df-bj-ind 10417 . 2 (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑧𝐴 suc 𝑧𝐴))
3516, 33, 34sylanbrc 402 1 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 639  wal 1257   = wceq 1259  wex 1397  wcel 1409  wral 2323  wrex 2324  c0 3251  suc csuc 4129  Ind wind 10416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910  ax-pr 3971  ax-un 4197  ax-bd0 10299  ax-bdor 10302  ax-bdex 10305  ax-bdeq 10306  ax-bdel 10307  ax-bdsb 10308  ax-bdsep 10370
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-nul 3252  df-sn 3408  df-pr 3409  df-uni 3608  df-suc 4135  df-bdc 10327  df-bj-ind 10417
This theorem is referenced by:  bj-inf2vn  10465  bj-inf2vn2  10466
  Copyright terms: Public domain W3C validator