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Theorem findset 10436
Description: Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4349 for a nonconstructive proof of the general case. See bdfind 10437 for a proof when 𝐴 is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
findset (𝐴𝑉 → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 = ω))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem findset
StepHypRef Expression
1 simpr1 921 . . 3 ((𝐴𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)) → 𝐴 ⊆ ω)
2 simp2 916 . . . . . 6 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → ∅ ∈ 𝐴)
3 df-ral 2328 . . . . . . . 8 (∀𝑥𝐴 suc 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴 → suc 𝑥𝐴))
4 alral 2384 . . . . . . . 8 (∀𝑥(𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
53, 4sylbi 118 . . . . . . 7 (∀𝑥𝐴 suc 𝑥𝐴 → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
653ad2ant3 938 . . . . . 6 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
72, 6jca 294 . . . . 5 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)))
8 3anass 900 . . . . . 6 ((𝐴𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ↔ (𝐴𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))))
98biimpri 128 . . . . 5 ((𝐴𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))) → (𝐴𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)))
107, 9sylan2 274 . . . 4 ((𝐴𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)) → (𝐴𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)))
11 speano5 10435 . . . 4 ((𝐴𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
1210, 11syl 14 . . 3 ((𝐴𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)) → ω ⊆ 𝐴)
131, 12eqssd 2989 . 2 ((𝐴𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)) → 𝐴 = ω)
1413ex 112 1 (𝐴𝑉 → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 = ω))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896  wal 1257   = wceq 1259  wcel 1409  wral 2323  wss 2944  c0 3251  suc csuc 4129  ωcom 4340
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-bdan 10301  ax-bdor 10302  ax-bdex 10305  ax-bdeq 10306  ax-bdel 10307  ax-bdsb 10308  ax-bdsep 10370  ax-infvn 10432
This theorem depends on definitions:  df-bi 114  df-3an 898  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-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-sn 3408  df-pr 3409  df-uni 3608  df-int 3643  df-suc 4135  df-iom 4341  df-bdc 10327  df-bj-ind 10417
This theorem is referenced by:  bdfind  10437
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