Theorem finds 4412

Description: Principle of finite induction over the finite cardinals, using implicit substitutions. The first hypothesis ensures stratification of φ, the next four set up the substitutions, and the last two set up the base case and induction hypothesis. Compare Theorem X.1.13 of [Rosser] p. 277. (Contributed by SF, 14-Jan-2015.)

Hypotheses

Ref	Expression
finds.1	⊢ {x ∣ φ} ∈ V
finds.2	⊢ (x = 0c → (φ ↔ ψ))
finds.3	⊢ (x = y → (φ ↔ χ))
finds.4	⊢ (x = (y +c 1c) → (φ ↔ θ))
finds.5	⊢ (x = A → (φ ↔ τ))
finds.6	⊢ ψ
finds.7	⊢ (y ∈ Nn → (χ → θ))

Assertion

Ref	Expression
finds	⊢ (A ∈ Nn → τ)

Distinct variable groups:   x,A   χ,x   φ,y   ψ,x   τ,x   θ,x   x,y

Allowed substitution hints:   φ(x)   ψ(y)   χ(y)   θ(y)   τ(y)   A(y)

Proof of Theorem finds

Step	Hyp	Ref	Expression
1		tru 1321	. 2 ⊢ ⊤
2		finds.1	. . . 4 ⊢ {x ∣ φ} ∈ V
3	2	a1i 10	. . 3 ⊢ ( ⊤ → {x ∣ φ} ∈ V)
4		finds.2	. . 3 ⊢ (x = 0c → (φ ↔ ψ))
5		finds.3	. . 3 ⊢ (x = y → (φ ↔ χ))
6		finds.4	. . 3 ⊢ (x = (y +c 1c) → (φ ↔ θ))
7		finds.5	. . 3 ⊢ (x = A → (φ ↔ τ))
8		finds.6	. . . 4 ⊢ ψ
9	8	a1i 10	. . 3 ⊢ ( ⊤ → ψ)
10		finds.7	. . . 4 ⊢ (y ∈ Nn → (χ → θ))
11	10	adantr 451	. . 3 ⊢ ((y ∈ Nn ∧ ⊤ ) → (χ → θ))
12	3, 4, 5, 6, 7, 9, 11	findsd 4411	. 2 ⊢ ((A ∈ Nn ∧ ⊤ ) → τ)
13	1, 12	mpan2 652	1 ⊢ (A ∈ Nn → τ)

Syntax hints:   → wi 4   ↔ wb 176   ⊤ wtru 1316   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-0c 4378  df-addc 4379  df-nnc 4380

This theorem is referenced by:  nnc0suc  4413  nncaddccl  4420  nnsucelr  4429  nndisjeq  4430  ltfintri  4467  ssfin  4471  ncfinraise  4482  ncfinlower  4484  evenoddnnnul  4515  evenodddisj  4517  nnadjoin  4521  nnpweq  4524  sfintfin  4533  tfinnn  4535  nulnnn  4557  nnnc  6147  ce0nn  6181  leconnnc  6219  nclenn  6250  nnltp1c  6263  nmembers1  6272  nncdiv3  6278  nnc3n3p1  6279  nchoicelem12  6301  nchoicelem17  6306