Theorem findsd 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. 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 4393	. . . . . . 7 ⊢ 0c ∈ V
4		findsd.2	. . . . . . 7 ⊢ (x = 0c → (φ ↔ ψ))
5	3, 4	elab 2986	. . . . . 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 2863	. . . . . . . . 9 ⊢ y ∈ V
9		findsd.3	. . . . . . . . 9 ⊢ (x = y → (φ ↔ χ))
10	8, 9	elab 2986	. . . . . . . 8 ⊢ (y ∈ {x ∣ φ} ↔ χ)
11		1cex 4143	. . . . . . . . . 10 ⊢ 1c ∈ V
12	8, 11	addcex 4395	. . . . . . . . 9 ⊢ (y +c 1c) ∈ V
13		findsd.4	. . . . . . . . 9 ⊢ (x = (y +c 1c) → (φ ↔ θ))
14	12, 13	elab 2986	. . . . . . . 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 2698	. . . . 5 ⊢ (η → ∀y ∈ Nn (y ∈ {x ∣ φ} → (y +c 1c) ∈ {x ∣ φ}))
18		peano5 4410	. . . . 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 3273	. . 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 2987	. . 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 2615   ⊆ wss 3258  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:  finds  4412  preaddccan2  4456  addccan2nc  6266  fnfrec  6321