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Mirrors > Home > MPE Home > Th. List > php4 | Structured version Visualization version GIF version |
Description: Corollary of the Pigeonhole Principle php 9228: 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 6444 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴) | |
2 | nnord 7872 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
3 | ordsuc 7810 | . . . . 5 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
4 | 3 | biimpi 215 | . . . 4 ⊢ (Ord 𝐴 → Ord suc 𝐴) |
5 | ordelpss 6391 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) | |
6 | 2, 4, 5 | syl2anc2 584 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) |
7 | 1, 6 | mpbid 231 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ⊊ suc 𝐴) |
8 | peano2b 7881 | . . 3 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | |
9 | php2 9229 | . . 3 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐴 ⊊ suc 𝐴) → 𝐴 ≺ suc 𝐴) | |
10 | 8, 9 | sylanb 580 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ⊊ suc 𝐴) → 𝐴 ≺ suc 𝐴) |
11 | 7, 10 | mpdan 686 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≺ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2099 ⊊ wpss 3946 class class class wbr 5142 Ord word 6362 suc csuc 6365 ωcom 7864 ≺ csdm 8956 |
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 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7865 df-1o 8480 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 |
This theorem is referenced by: php5 9232 sucdom 9253 sucdomOLD 9254 1sdom2ALT 9259 domalom 36877 |
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