Theorem bdfind 14549

Description: Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4597 for a nonconstructive proof of the general case. See findset 14548 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 14545	. . . 4 ⊢ ω ∈ V
3	1, 2	bdssex 14505	. . 3 ⊢ (𝐴 ⊆ ω → 𝐴 ∈ V)
4	3	3ad2ant1 1018	. 2 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
5		findset 14548	. 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 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2737   ⊆ wss 3129  ∅c0 3422  suc csuc 4364  ωcom 4588  BOUNDED wbdc 14443

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4128  ax-pr 4208  ax-un 4432  ax-bd0 14416  ax-bdan 14418  ax-bdor 14419  ax-bdex 14422  ax-bdeq 14423  ax-bdel 14424  ax-bdsb 14425  ax-bdsep 14487  ax-infvn 14544

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4370  df-iom 4589  df-bdc 14444  df-bj-ind 14530

This theorem is referenced by: (None)