Theorem bdfind 16686

Description: Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4712 for a nonconstructive proof of the general case. See findset 16685 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 16682	. . . 4 ⊢ ω ∈ V
3	1, 2	bdssex 16642	. . 3 ⊢ (𝐴 ⊆ ω → 𝐴 ∈ V)
4	3	3ad2ant1 1045	. 2 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 ∈ V)
5		findset 16685	. 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 1005   = wceq 1398   ∈ wcel 2203  ∀wral 2520  Vcvv 2812   ⊆ wss 3210  ∅c0 3505  suc csuc 4477  ωcom 4703  BOUNDED wbdc 16580

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-nul 4229  ax-pr 4314  ax-un 4545  ax-bd0 16553  ax-bdan 16555  ax-bdor 16556  ax-bdex 16559  ax-bdeq 16560  ax-bdel 16561  ax-bdsb 16562  ax-bdsep 16624  ax-infvn 16681

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-sn 3688  df-pr 3689  df-uni 3908  df-int 3943  df-suc 4483  df-iom 4704  df-bdc 16581  df-bj-ind 16667

This theorem is referenced by: (None)