New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > finds | GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
finds | ⊢ (A ∈ Nn → τ) |
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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ⊤ wtru 1316 = wceq 1642 ∈ wcel 1710 {cab 2339 Vcvv 2859 1cc1c 4134 Nn cnnc 4373 0cc0c 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 |
Copyright terms: Public domain | W3C validator |