![]() |
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 9235: 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 6447 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴) | |
2 | nnord 7874 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
3 | ordsuc 7812 | . . . . 5 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
4 | 3 | biimpi 215 | . . . 4 ⊢ (Ord 𝐴 → Ord suc 𝐴) |
5 | ordelpss 6394 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) | |
6 | 2, 4, 5 | syl2anc2 583 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) |
7 | 1, 6 | mpbid 231 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ⊊ suc 𝐴) |
8 | peano2b 7883 | . . 3 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | |
9 | php2 9236 | . . 3 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐴 ⊊ suc 𝐴) → 𝐴 ≺ suc 𝐴) | |
10 | 8, 9 | sylanb 579 | . 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 2099 ⊊ wpss 3948 class class class wbr 5144 Ord word 6365 suc csuc 6368 ωcom 7866 ≺ csdm 8963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-om 7867 df-1o 8486 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 |
This theorem is referenced by: php5 9239 sucdom 9260 sucdomOLD 9261 1sdom2ALT 9266 domalom 37122 |
Copyright terms: Public domain | W3C validator |