![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > php4 | Structured version Visualization version GIF version |
Description: Corollary of the Pigeonhole Principle php 9206: a natural number is strictly dominated by its successor. (Contributed by NM, 26-Jul-2004.) |
Ref | Expression |
---|---|
php4 | ⊢ (𝐴 ∈ ω → 𝐴 ≺ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucidg 6442 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴) | |
2 | nnord 7859 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
3 | ordsuc 7797 | . . . . 5 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
4 | 3 | biimpi 215 | . . . 4 ⊢ (Ord 𝐴 → Ord suc 𝐴) |
5 | ordelpss 6389 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) | |
6 | 2, 4, 5 | syl2anc2 585 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) |
7 | 1, 6 | mpbid 231 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ⊊ suc 𝐴) |
8 | peano2b 7868 | . . 3 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | |
9 | php2 9207 | . . 3 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐴 ⊊ suc 𝐴) → 𝐴 ≺ suc 𝐴) | |
10 | 8, 9 | sylanb 581 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ⊊ suc 𝐴) → 𝐴 ≺ suc 𝐴) |
11 | 7, 10 | mpdan 685 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≺ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 ⊊ wpss 3948 class class class wbr 5147 Ord word 6360 suc csuc 6363 ωcom 7851 ≺ csdm 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7852 df-1o 8462 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 |
This theorem is referenced by: php5 9210 sucdom 9231 sucdomOLD 9232 1sdom2ALT 9237 domalom 36273 |
Copyright terms: Public domain | W3C validator |