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 8689: 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 6262 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴) | |
2 | nnord 7577 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
3 | ordsuc 7518 | . . . . 5 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
4 | 3 | biimpi 217 | . . . 4 ⊢ (Ord 𝐴 → Ord suc 𝐴) |
5 | ordelpss 6212 | . . . 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 233 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ⊊ suc 𝐴) |
8 | peano2b 7585 | . . 3 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | |
9 | php2 8690 | . . 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 683 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≺ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 ⊊ wpss 3934 class class class wbr 5057 Ord word 6183 suc csuc 6186 ωcom 7569 ≺ csdm 8496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 |
This theorem is referenced by: php5 8693 sucdom 8703 1sdom2 8705 domalom 34567 |
Copyright terms: Public domain | W3C validator |