Theorem bdfind 16267

Description: Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4690 for a nonconstructive proof of the general case. See findset 16266 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 16263	. . . 4 ⊢ ω ∈ V
3	1, 2	bdssex 16223	. . 3 ⊢ (𝐴 ⊆ ω → 𝐴 ∈ V)
4	3	3ad2ant1 1042	. 2 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
5		findset 16266	. 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 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ⊆ wss 3197  ∅c0 3491  suc csuc 4455  ωcom 4681  BOUNDED wbdc 16161

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-nul 4209  ax-pr 4292  ax-un 4523  ax-bd0 16134  ax-bdan 16136  ax-bdor 16137  ax-bdex 16140  ax-bdeq 16141  ax-bdel 16142  ax-bdsb 16143  ax-bdsep 16205  ax-infvn 16262

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-pr 3673  df-uni 3888  df-int 3923  df-suc 4461  df-iom 4682  df-bdc 16162  df-bj-ind 16248

This theorem is referenced by: (None)