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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 4387 | . . . 4 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅) | |
3 | 1, 2 | breq12d 4002 | . . 3 ⊢ (𝑤 = ∅ → (𝑤 ≈ suc 𝑤 ↔ ∅ ≈ suc ∅)) |
4 | 3 | notbid 662 | . 2 ⊢ (𝑤 = ∅ → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ ∅ ≈ suc ∅)) |
5 | id 19 | . . . 4 ⊢ (𝑤 = 𝑘 → 𝑤 = 𝑘) | |
6 | suceq 4387 | . . . 4 ⊢ (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘) | |
7 | 5, 6 | breq12d 4002 | . . 3 ⊢ (𝑤 = 𝑘 → (𝑤 ≈ suc 𝑤 ↔ 𝑘 ≈ suc 𝑘)) |
8 | 7 | notbid 662 | . 2 ⊢ (𝑤 = 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝑘 ≈ suc 𝑘)) |
9 | id 19 | . . . 4 ⊢ (𝑤 = suc 𝑘 → 𝑤 = suc 𝑘) | |
10 | suceq 4387 | . . . 4 ⊢ (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘) | |
11 | 9, 10 | breq12d 4002 | . . 3 ⊢ (𝑤 = suc 𝑘 → (𝑤 ≈ suc 𝑤 ↔ suc 𝑘 ≈ suc suc 𝑘)) |
12 | 11 | notbid 662 | . 2 ⊢ (𝑤 = suc 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ suc 𝑘 ≈ suc suc 𝑘)) |
13 | id 19 | . . . 4 ⊢ (𝑤 = 𝐴 → 𝑤 = 𝐴) | |
14 | suceq 4387 | . . . 4 ⊢ (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴) | |
15 | 13, 14 | breq12d 4002 | . . 3 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ suc 𝑤 ↔ 𝐴 ≈ suc 𝐴)) |
16 | 15 | notbid 662 | . 2 ⊢ (𝑤 = 𝐴 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝐴 ≈ suc 𝐴)) |
17 | peano1 4578 | . . . . 5 ⊢ ∅ ∈ ω | |
18 | peano3 4580 | . . . . 5 ⊢ (∅ ∈ ω → suc ∅ ≠ ∅) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ suc ∅ ≠ ∅ |
20 | en0 6773 | . . . 4 ⊢ (suc ∅ ≈ ∅ ↔ suc ∅ = ∅) | |
21 | 19, 20 | nemtbir 2429 | . . 3 ⊢ ¬ suc ∅ ≈ ∅ |
22 | ensymb 6758 | . . 3 ⊢ (suc ∅ ≈ ∅ ↔ ∅ ≈ suc ∅) | |
23 | 21, 22 | mtbi 665 | . 2 ⊢ ¬ ∅ ≈ suc ∅ |
24 | peano2 4579 | . . . 4 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω) | |
25 | vex 2733 | . . . . 5 ⊢ 𝑘 ∈ V | |
26 | 25 | sucex 4483 | . . . . 5 ⊢ suc 𝑘 ∈ V |
27 | 25, 26 | phplem4 6833 | . . . 4 ⊢ ((𝑘 ∈ ω ∧ suc 𝑘 ∈ ω) → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘)) |
28 | 24, 27 | mpdan 419 | . . 3 ⊢ (𝑘 ∈ ω → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘)) |
29 | 28 | con3d 626 | . 2 ⊢ (𝑘 ∈ ω → (¬ 𝑘 ≈ suc 𝑘 → ¬ suc 𝑘 ≈ suc suc 𝑘)) |
30 | 4, 8, 12, 16, 23, 29 | finds 4584 | 1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∅c0 3414 class class class wbr 3989 suc csuc 4350 ωcom 4574 ≈ cen 6716 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-er 6513 df-en 6719 |
This theorem is referenced by: snnen2og 6837 1nen2 6839 php5dom 6841 php5fin 6860 |
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