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Mirrors > Home > NFE Home > Th. List > sfin01 | GIF version |
Description: Zero and one satisfy Sfin. Theorem X.1.42 of [Rosser] p. 530. (Contributed by SF, 30-Jan-2015.) |
Ref | Expression |
---|---|
sfin01 | ⊢ Sfin (0c, 1c) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 4402 | . 2 ⊢ 0c ∈ Nn | |
2 | 1cnnc 4408 | . 2 ⊢ 1c ∈ Nn | |
3 | pw10 4161 | . . 3 ⊢ ℘1∅ = ∅ | |
4 | 0ex 4110 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | snel1c 4140 | . . 3 ⊢ {∅} ∈ 1c |
6 | el0c 4421 | . . . . . 6 ⊢ (℘1a ∈ 0c ↔ ℘1a = ∅) | |
7 | pw1eq 4143 | . . . . . . 7 ⊢ (a = ∅ → ℘1a = ℘1∅) | |
8 | 7 | eqeq1d 2361 | . . . . . 6 ⊢ (a = ∅ → (℘1a = ∅ ↔ ℘1∅ = ∅)) |
9 | 6, 8 | syl5bb 248 | . . . . 5 ⊢ (a = ∅ → (℘1a ∈ 0c ↔ ℘1∅ = ∅)) |
10 | pweq 3725 | . . . . . . 7 ⊢ (a = ∅ → ℘a = ℘∅) | |
11 | pw0 4160 | . . . . . . 7 ⊢ ℘∅ = {∅} | |
12 | 10, 11 | syl6eq 2401 | . . . . . 6 ⊢ (a = ∅ → ℘a = {∅}) |
13 | 12 | eleq1d 2419 | . . . . 5 ⊢ (a = ∅ → (℘a ∈ 1c ↔ {∅} ∈ 1c)) |
14 | 9, 13 | anbi12d 691 | . . . 4 ⊢ (a = ∅ → ((℘1a ∈ 0c ∧ ℘a ∈ 1c) ↔ (℘1∅ = ∅ ∧ {∅} ∈ 1c))) |
15 | 4, 14 | spcev 2946 | . . 3 ⊢ ((℘1∅ = ∅ ∧ {∅} ∈ 1c) → ∃a(℘1a ∈ 0c ∧ ℘a ∈ 1c)) |
16 | 3, 5, 15 | mp2an 653 | . 2 ⊢ ∃a(℘1a ∈ 0c ∧ ℘a ∈ 1c) |
17 | df-sfin 4446 | . 2 ⊢ ( Sfin (0c, 1c) ↔ (0c ∈ Nn ∧ 1c ∈ Nn ∧ ∃a(℘1a ∈ 0c ∧ ℘a ∈ 1c))) | |
18 | 1, 2, 16, 17 | mpbir3an 1134 | 1 ⊢ Sfin (0c, 1c) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∅c0 3550 ℘cpw 3722 {csn 3737 1cc1c 4134 ℘1cpw1 4135 Nn cnnc 4373 0cc0c 4374 Sfin wsfin 4438 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-pw 3724 df-sn 3741 df-pr 3742 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-p6 4191 df-sik 4192 df-ssetk 4193 df-0c 4377 df-addc 4378 df-nnc 4379 df-sfin 4446 |
This theorem is referenced by: sfintfin 4532 |
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