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Theorem findset 13070
Description: Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4483 for a nonconstructive proof of the general case. See bdfind 13071 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 972 . . 3 ((𝐴𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)) → 𝐴 ⊆ ω)
2 simp2 967 . . . . . 6 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → ∅ ∈ 𝐴)
3 df-ral 2398 . . . . . . . 8 (∀𝑥𝐴 suc 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴 → suc 𝑥𝐴))
4 alral 2455 . . . . . . . 8 (∀𝑥(𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
53, 4sylbi 120 . . . . . . 7 (∀𝑥𝐴 suc 𝑥𝐴 → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
653ad2ant3 989 . . . . . 6 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))
72, 6jca 304 . . . . 5 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)))
8 3anass 951 . . . . . 6 ((𝐴𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ↔ (𝐴𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))))
98biimpri 132 . . . . 5 ((𝐴𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))) → (𝐴𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)))
107, 9sylan2 284 . . . 4 ((𝐴𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)) → (𝐴𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)))
11 speano5 13069 . . . 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 3084 . 2 ((𝐴𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)) → 𝐴 = ω)
1413ex 114 1 (𝐴𝑉 → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 = ω))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947  wal 1314   = wceq 1316  wcel 1465  wral 2393  wss 3041  c0 3333  suc csuc 4257  ωcom 4474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024  ax-pr 4101  ax-un 4325  ax-bd0 12938  ax-bdan 12940  ax-bdor 12941  ax-bdex 12944  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947  ax-bdsep 13009  ax-infvn 13066
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475  df-bdc 12966  df-bj-ind 13052
This theorem is referenced by:  bdfind  13071
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