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Mirrors > Home > NFE Home > Th. List > peano1 | GIF version |
Description: Cardinal zero is a finite cardinal. Theorem X.1.4 of [Rosser] p. 276. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
peano1 | ⊢ 0c ∈ Nn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nnc 4379 | . . . 4 ⊢ Nn = ∩{x ∣ (0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x)} | |
2 | 1 | eleq2i 2417 | . . 3 ⊢ (0c ∈ Nn ↔ 0c ∈ ∩{x ∣ (0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x)}) |
3 | 0cex 4392 | . . . 4 ⊢ 0c ∈ V | |
4 | 3 | elintab 3937 | . . 3 ⊢ (0c ∈ ∩{x ∣ (0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x)} ↔ ∀x((0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x) → 0c ∈ x)) |
5 | 2, 4 | bitri 240 | . 2 ⊢ (0c ∈ Nn ↔ ∀x((0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x) → 0c ∈ x)) |
6 | simpl 443 | . 2 ⊢ ((0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x) → 0c ∈ x) | |
7 | 5, 6 | mpgbir 1550 | 1 ⊢ 0c ∈ Nn |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∀wal 1540 ∈ wcel 1710 {cab 2339 ∀wral 2614 ∩cint 3926 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-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-int 3927 df-0c 4377 df-nnc 4379 |
This theorem is referenced by: 1cnnc 4408 peano5 4409 nnc0suc 4412 0fin 4423 ltfinirr 4457 0cminle 4461 ltfinp1 4462 lefinlteq 4463 lefinrflx 4467 ncfinraise 4481 ncfinlower 4483 tfin0c 4497 tfin1c 4499 0ceven 4505 sfin01 4528 vfin1cltv 4547 0cnelphi 4597 ncssfin 6151 nclenn 6249 nncdiv3 6277 nnc3n3p1 6278 nchoicelem17 6305 frecxp 6314 frec0 6321 |
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