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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 4427 | . . . 4 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅) | |
3 | 1, 2 | breq12d 4038 | . . 3 ⊢ (𝑤 = ∅ → (𝑤 ≈ suc 𝑤 ↔ ∅ ≈ suc ∅)) |
4 | 3 | notbid 668 | . 2 ⊢ (𝑤 = ∅ → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ ∅ ≈ suc ∅)) |
5 | id 19 | . . . 4 ⊢ (𝑤 = 𝑘 → 𝑤 = 𝑘) | |
6 | suceq 4427 | . . . 4 ⊢ (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘) | |
7 | 5, 6 | breq12d 4038 | . . 3 ⊢ (𝑤 = 𝑘 → (𝑤 ≈ suc 𝑤 ↔ 𝑘 ≈ suc 𝑘)) |
8 | 7 | notbid 668 | . 2 ⊢ (𝑤 = 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝑘 ≈ suc 𝑘)) |
9 | id 19 | . . . 4 ⊢ (𝑤 = suc 𝑘 → 𝑤 = suc 𝑘) | |
10 | suceq 4427 | . . . 4 ⊢ (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘) | |
11 | 9, 10 | breq12d 4038 | . . 3 ⊢ (𝑤 = suc 𝑘 → (𝑤 ≈ suc 𝑤 ↔ suc 𝑘 ≈ suc suc 𝑘)) |
12 | 11 | notbid 668 | . 2 ⊢ (𝑤 = suc 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ suc 𝑘 ≈ suc suc 𝑘)) |
13 | id 19 | . . . 4 ⊢ (𝑤 = 𝐴 → 𝑤 = 𝐴) | |
14 | suceq 4427 | . . . 4 ⊢ (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴) | |
15 | 13, 14 | breq12d 4038 | . . 3 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ suc 𝑤 ↔ 𝐴 ≈ suc 𝐴)) |
16 | 15 | notbid 668 | . 2 ⊢ (𝑤 = 𝐴 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝐴 ≈ suc 𝐴)) |
17 | peano1 4618 | . . . . 5 ⊢ ∅ ∈ ω | |
18 | peano3 4620 | . . . . 5 ⊢ (∅ ∈ ω → suc ∅ ≠ ∅) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ suc ∅ ≠ ∅ |
20 | en0 6836 | . . . 4 ⊢ (suc ∅ ≈ ∅ ↔ suc ∅ = ∅) | |
21 | 19, 20 | nemtbir 2449 | . . 3 ⊢ ¬ suc ∅ ≈ ∅ |
22 | ensymb 6821 | . . 3 ⊢ (suc ∅ ≈ ∅ ↔ ∅ ≈ suc ∅) | |
23 | 21, 22 | mtbi 671 | . 2 ⊢ ¬ ∅ ≈ suc ∅ |
24 | peano2 4619 | . . . 4 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω) | |
25 | vex 2759 | . . . . 5 ⊢ 𝑘 ∈ V | |
26 | 25 | sucex 4523 | . . . . 5 ⊢ suc 𝑘 ∈ V |
27 | 25, 26 | phplem4 6898 | . . . 4 ⊢ ((𝑘 ∈ ω ∧ suc 𝑘 ∈ ω) → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘)) |
28 | 24, 27 | mpdan 421 | . . 3 ⊢ (𝑘 ∈ ω → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘)) |
29 | 28 | con3d 632 | . 2 ⊢ (𝑘 ∈ ω → (¬ 𝑘 ≈ suc 𝑘 → ¬ suc 𝑘 ≈ suc suc 𝑘)) |
30 | 4, 8, 12, 16, 23, 29 | finds 4624 | 1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∅c0 3442 class class class wbr 4025 suc csuc 4390 ωcom 4614 ≈ cen 6779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2758 df-sbc 2982 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-br 4026 df-opab 4087 df-tr 4124 df-id 4318 df-iord 4391 df-on 4393 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-er 6574 df-en 6782 |
This theorem is referenced by: snnen2og 6902 1nen2 6904 php5dom 6906 php5fin 6925 |
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