Theorem findsd 4410

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. This version allows for an extra deduction clause that may make proving stratification simpler. Compare Theorem X.1.13 of [Rosser] p. 277. (Contributed by SF, 31-Jul-2019.)

Hypotheses

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

Assertion

Ref	Expression
findsd	⊢ ((A ∈ Nn ∧ η) → τ)

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

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

Proof of Theorem findsd

Step	Hyp	Ref	Expression
1		findsd.1	. . . . 5 ⊢ (η → {x ∣ φ} ∈ V)
2		findsd.6	. . . . . 6 ⊢ (η → ψ)
3		0cex 4392	. . . . . . 7 ⊢ 0c ∈ V
4		findsd.2	. . . . . . 7 ⊢ (x = 0c → (φ ↔ ψ))
5	3, 4	elab 2985	. . . . . 6 ⊢ (0c ∈ {x ∣ φ} ↔ ψ)
6	2, 5	sylibr 203	. . . . 5 ⊢ (η → 0c ∈ {x ∣ φ})
7		findsd.7	. . . . . . . 8 ⊢ ((y ∈ Nn ∧ η) → (χ → θ))
8		vex 2862	. . . . . . . . 9 ⊢ y ∈ V
9		findsd.3	. . . . . . . . 9 ⊢ (x = y → (φ ↔ χ))
10	8, 9	elab 2985	. . . . . . . 8 ⊢ (y ∈ {x ∣ φ} ↔ χ)
11		1cex 4142	. . . . . . . . . 10 ⊢ 1c ∈ V
12	8, 11	addcex 4394	. . . . . . . . 9 ⊢ (y +c 1c) ∈ V
13		findsd.4	. . . . . . . . 9 ⊢ (x = (y +c 1c) → (φ ↔ θ))
14	12, 13	elab 2985	. . . . . . . 8 ⊢ ((y +c 1c) ∈ {x ∣ φ} ↔ θ)
15	7, 10, 14	3imtr4g 261	. . . . . . 7 ⊢ ((y ∈ Nn ∧ η) → (y ∈ {x ∣ φ} → (y +c 1c) ∈ {x ∣ φ}))
16	15	ancoms 439	. . . . . 6 ⊢ ((η ∧ y ∈ Nn ) → (y ∈ {x ∣ φ} → (y +c 1c) ∈ {x ∣ φ}))
17	16	ralrimiva 2697	. . . . 5 ⊢ (η → ∀y ∈ Nn (y ∈ {x ∣ φ} → (y +c 1c) ∈ {x ∣ φ}))
18		peano5 4409	. . . . 5 ⊢ (({x ∣ φ} ∈ V ∧ 0c ∈ {x ∣ φ} ∧ ∀y ∈ Nn (y ∈ {x ∣ φ} → (y +c 1c) ∈ {x ∣ φ})) → Nn ⊆ {x ∣ φ})
19	1, 6, 17, 18	syl3anc 1182	. . . 4 ⊢ (η → Nn ⊆ {x ∣ φ})
20	19	sseld 3272	. . 3 ⊢ (η → (A ∈ Nn → A ∈ {x ∣ φ}))
21	20	impcom 419	. 2 ⊢ ((A ∈ Nn ∧ η) → A ∈ {x ∣ φ})
22		findsd.5	. . . 4 ⊢ (x = A → (φ ↔ τ))
23	22	elabg 2986	. . 3 ⊢ (A ∈ Nn → (A ∈ {x ∣ φ} ↔ τ))
24	23	adantr 451	. 2 ⊢ ((A ∈ Nn ∧ η) → (A ∈ {x ∣ φ} ↔ τ))
25	21, 24	mpbid 201	1 ⊢ ((A ∈ Nn ∧ η) → τ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614   ⊆ wss 3257  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:  finds  4411  preaddccan2  4455  addccan2nc  6265  fnfrec  6320