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Mirrors > Home > MPE Home > Th. List > php4 | Structured version Visualization version GIF version |
Description: Corollary of the Pigeonhole Principle php 8413: 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 6041 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴) | |
2 | nnord 7334 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
3 | ordsuc 7275 | . . . . . 6 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
4 | 3 | biimpi 208 | . . . . 5 ⊢ (Ord 𝐴 → Ord suc 𝐴) |
5 | 4 | ancli 544 | . . . 4 ⊢ (Ord 𝐴 → (Ord 𝐴 ∧ Ord suc 𝐴)) |
6 | ordelpss 5991 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) | |
7 | 2, 5, 6 | 3syl 18 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) |
8 | 1, 7 | mpbid 224 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ⊊ suc 𝐴) |
9 | peano2b 7342 | . . 3 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | |
10 | php2 8414 | . . 3 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐴 ⊊ suc 𝐴) → 𝐴 ≺ suc 𝐴) | |
11 | 9, 10 | sylanb 576 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ⊊ suc 𝐴) → 𝐴 ≺ suc 𝐴) |
12 | 8, 11 | mpdan 678 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≺ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2164 ⊊ wpss 3799 class class class wbr 4873 Ord word 5962 suc csuc 5965 ωcom 7326 ≺ csdm 8221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-om 7327 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 |
This theorem is referenced by: php5 8417 sucdom 8426 1sdom2 8428 |
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