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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 4374 | . . . 4 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅) | |
3 | 1, 2 | breq12d 3989 | . . 3 ⊢ (𝑤 = ∅ → (𝑤 ≈ suc 𝑤 ↔ ∅ ≈ suc ∅)) |
4 | 3 | notbid 657 | . 2 ⊢ (𝑤 = ∅ → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ ∅ ≈ suc ∅)) |
5 | id 19 | . . . 4 ⊢ (𝑤 = 𝑘 → 𝑤 = 𝑘) | |
6 | suceq 4374 | . . . 4 ⊢ (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘) | |
7 | 5, 6 | breq12d 3989 | . . 3 ⊢ (𝑤 = 𝑘 → (𝑤 ≈ suc 𝑤 ↔ 𝑘 ≈ suc 𝑘)) |
8 | 7 | notbid 657 | . 2 ⊢ (𝑤 = 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝑘 ≈ suc 𝑘)) |
9 | id 19 | . . . 4 ⊢ (𝑤 = suc 𝑘 → 𝑤 = suc 𝑘) | |
10 | suceq 4374 | . . . 4 ⊢ (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘) | |
11 | 9, 10 | breq12d 3989 | . . 3 ⊢ (𝑤 = suc 𝑘 → (𝑤 ≈ suc 𝑤 ↔ suc 𝑘 ≈ suc suc 𝑘)) |
12 | 11 | notbid 657 | . 2 ⊢ (𝑤 = suc 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ suc 𝑘 ≈ suc suc 𝑘)) |
13 | id 19 | . . . 4 ⊢ (𝑤 = 𝐴 → 𝑤 = 𝐴) | |
14 | suceq 4374 | . . . 4 ⊢ (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴) | |
15 | 13, 14 | breq12d 3989 | . . 3 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ suc 𝑤 ↔ 𝐴 ≈ suc 𝐴)) |
16 | 15 | notbid 657 | . 2 ⊢ (𝑤 = 𝐴 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝐴 ≈ suc 𝐴)) |
17 | peano1 4565 | . . . . 5 ⊢ ∅ ∈ ω | |
18 | peano3 4567 | . . . . 5 ⊢ (∅ ∈ ω → suc ∅ ≠ ∅) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ suc ∅ ≠ ∅ |
20 | en0 6752 | . . . 4 ⊢ (suc ∅ ≈ ∅ ↔ suc ∅ = ∅) | |
21 | 19, 20 | nemtbir 2423 | . . 3 ⊢ ¬ suc ∅ ≈ ∅ |
22 | ensymb 6737 | . . 3 ⊢ (suc ∅ ≈ ∅ ↔ ∅ ≈ suc ∅) | |
23 | 21, 22 | mtbi 660 | . 2 ⊢ ¬ ∅ ≈ suc ∅ |
24 | peano2 4566 | . . . 4 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω) | |
25 | vex 2724 | . . . . 5 ⊢ 𝑘 ∈ V | |
26 | 25 | sucex 4470 | . . . . 5 ⊢ suc 𝑘 ∈ V |
27 | 25, 26 | phplem4 6812 | . . . 4 ⊢ ((𝑘 ∈ ω ∧ suc 𝑘 ∈ ω) → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘)) |
28 | 24, 27 | mpdan 418 | . . 3 ⊢ (𝑘 ∈ ω → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘)) |
29 | 28 | con3d 621 | . 2 ⊢ (𝑘 ∈ ω → (¬ 𝑘 ≈ suc 𝑘 → ¬ suc 𝑘 ≈ suc suc 𝑘)) |
30 | 4, 8, 12, 16, 23, 29 | finds 4571 | 1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 ∅c0 3404 class class class wbr 3976 suc csuc 4337 ωcom 4561 ≈ cen 6695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-er 6492 df-en 6698 |
This theorem is referenced by: snnen2og 6816 1nen2 6818 php5dom 6820 php5fin 6839 |
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