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Mirrors > Home > ILE Home > Th. List > php5 | GIF version |
Description: A natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.) |
Ref | Expression |
---|---|
php5 | ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . 4 ⊢ (𝑤 = ∅ → 𝑤 = ∅) | |
2 | suceq 4139 | . . . 4 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅) | |
3 | 1, 2 | breq12d 3777 | . . 3 ⊢ (𝑤 = ∅ → (𝑤 ≈ suc 𝑤 ↔ ∅ ≈ suc ∅)) |
4 | 3 | notbid 592 | . 2 ⊢ (𝑤 = ∅ → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ ∅ ≈ suc ∅)) |
5 | id 19 | . . . 4 ⊢ (𝑤 = 𝑘 → 𝑤 = 𝑘) | |
6 | suceq 4139 | . . . 4 ⊢ (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘) | |
7 | 5, 6 | breq12d 3777 | . . 3 ⊢ (𝑤 = 𝑘 → (𝑤 ≈ suc 𝑤 ↔ 𝑘 ≈ suc 𝑘)) |
8 | 7 | notbid 592 | . 2 ⊢ (𝑤 = 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝑘 ≈ suc 𝑘)) |
9 | id 19 | . . . 4 ⊢ (𝑤 = suc 𝑘 → 𝑤 = suc 𝑘) | |
10 | suceq 4139 | . . . 4 ⊢ (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘) | |
11 | 9, 10 | breq12d 3777 | . . 3 ⊢ (𝑤 = suc 𝑘 → (𝑤 ≈ suc 𝑤 ↔ suc 𝑘 ≈ suc suc 𝑘)) |
12 | 11 | notbid 592 | . 2 ⊢ (𝑤 = suc 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ suc 𝑘 ≈ suc suc 𝑘)) |
13 | id 19 | . . . 4 ⊢ (𝑤 = 𝐴 → 𝑤 = 𝐴) | |
14 | suceq 4139 | . . . 4 ⊢ (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴) | |
15 | 13, 14 | breq12d 3777 | . . 3 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ suc 𝑤 ↔ 𝐴 ≈ suc 𝐴)) |
16 | 15 | notbid 592 | . 2 ⊢ (𝑤 = 𝐴 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝐴 ≈ suc 𝐴)) |
17 | peano1 4317 | . . . . 5 ⊢ ∅ ∈ ω | |
18 | peano3 4319 | . . . . 5 ⊢ (∅ ∈ ω → suc ∅ ≠ ∅) | |
19 | 17, 18 | ax-mp 7 | . . . 4 ⊢ suc ∅ ≠ ∅ |
20 | en0 6275 | . . . 4 ⊢ (suc ∅ ≈ ∅ ↔ suc ∅ = ∅) | |
21 | 19, 20 | nemtbir 2294 | . . 3 ⊢ ¬ suc ∅ ≈ ∅ |
22 | ensymb 6260 | . . 3 ⊢ (suc ∅ ≈ ∅ ↔ ∅ ≈ suc ∅) | |
23 | 21, 22 | mtbi 595 | . 2 ⊢ ¬ ∅ ≈ suc ∅ |
24 | peano2 4318 | . . . 4 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω) | |
25 | vex 2560 | . . . . 5 ⊢ 𝑘 ∈ V | |
26 | 25 | sucex 4225 | . . . . 5 ⊢ suc 𝑘 ∈ V |
27 | 25, 26 | phplem4 6318 | . . . 4 ⊢ ((𝑘 ∈ ω ∧ suc 𝑘 ∈ ω) → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘)) |
28 | 24, 27 | mpdan 398 | . . 3 ⊢ (𝑘 ∈ ω → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘)) |
29 | 28 | con3d 561 | . 2 ⊢ (𝑘 ∈ ω → (¬ 𝑘 ≈ suc 𝑘 → ¬ suc 𝑘 ≈ suc suc 𝑘)) |
30 | 4, 8, 12, 16, 23, 29 | finds 4323 | 1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1243 ∈ wcel 1393 ≠ wne 2204 ∅c0 3224 class class class wbr 3764 suc csuc 4102 ωcom 4313 ≈ cen 6219 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-er 6106 df-en 6222 |
This theorem is referenced by: snnen2og 6322 php5dom 6325 php5fin 6339 |
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