Theorem bdfind 15508

Description: Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4632 for a nonconstructive proof of the general case. See findset 15507 for a proof when 𝐴 is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bdfind.bd	⊢ BOUNDED 𝐴

Assertion

Ref	Expression
bdfind	⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω)

Distinct variable group:   𝑥,𝐴

Proof of Theorem bdfind

Step	Hyp	Ref	Expression
1		bdfind.bd	. . . 4 ⊢ BOUNDED 𝐴
2		bj-omex 15504	. . . 4 ⊢ ω ∈ V
3	1, 2	bdssex 15464	. . 3 ⊢ (𝐴 ⊆ ω → 𝐴 ∈ V)
4	3	3ad2ant1 1020	. 2 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
5		findset 15507	. 2 ⊢ (𝐴 ∈ V → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω))
6	4, 5	mpcom 36	1 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω)

Syntax hints:   → wi 4   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472  Vcvv 2760   ⊆ wss 3154  ∅c0 3447  suc csuc 4397  ωcom 4623  BOUNDED wbdc 15402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-nul 4156  ax-pr 4239  ax-un 4465  ax-bd0 15375  ax-bdan 15377  ax-bdor 15378  ax-bdex 15381  ax-bdeq 15382  ax-bdel 15383  ax-bdsb 15384  ax-bdsep 15446  ax-infvn 15503

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-sn 3625  df-pr 3626  df-uni 3837  df-int 3872  df-suc 4403  df-iom 4624  df-bdc 15403  df-bj-ind 15489

This theorem is referenced by: (None)