Theorem finds 4411

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 4410	. 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 2859  1_cc1c 4134   Nn cnnc 4373  0_cc0c 4374   +_c cplc 4375

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-0c 4377  df-addc 4378  df-nnc 4379

This theorem is referenced by:  nnc0suc  4412  nncaddccl  4419  nnsucelr  4428  nndisjeq  4429  ltfintri  4466  ssfin  4470  ncfinraise  4481  ncfinlower  4483  evenoddnnnul  4514  evenodddisj  4516  nnadjoin  4520  nnpweq  4523  sfintfin  4532  tfinnn  4534  nulnnn  4556  nnnc  6146  ce0nn  6180  leconnnc  6218  nclenn  6249  nnltp1c  6262  nmembers1  6271  nncdiv3  6277  nnc3n3p1  6278  nchoicelem12  6300  nchoicelem17  6305