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Mirrors > Home > MPE Home > Th. List > 1lt2pi | Structured version Visualization version GIF version |
Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1lt2pi | ⊢ 1𝑜 <N (1𝑜 +N 1𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 7606 | . . . . 5 ⊢ 1𝑜 ∈ ω | |
2 | nna0 7571 | . . . . 5 ⊢ (1𝑜 ∈ ω → (1𝑜 +𝑜 ∅) = 1𝑜) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1𝑜 +𝑜 ∅) = 1𝑜 |
4 | 0lt1o 7471 | . . . . 5 ⊢ ∅ ∈ 1𝑜 | |
5 | peano1 6977 | . . . . . 6 ⊢ ∅ ∈ ω | |
6 | nnaord 7586 | . . . . . 6 ⊢ ((∅ ∈ ω ∧ 1𝑜 ∈ ω ∧ 1𝑜 ∈ ω) → (∅ ∈ 1𝑜 ↔ (1𝑜 +𝑜 ∅) ∈ (1𝑜 +𝑜 1𝑜))) | |
7 | 5, 1, 1, 6 | mp3an 1416 | . . . . 5 ⊢ (∅ ∈ 1𝑜 ↔ (1𝑜 +𝑜 ∅) ∈ (1𝑜 +𝑜 1𝑜)) |
8 | 4, 7 | mpbi 219 | . . . 4 ⊢ (1𝑜 +𝑜 ∅) ∈ (1𝑜 +𝑜 1𝑜) |
9 | 3, 8 | eqeltrri 2685 | . . 3 ⊢ 1𝑜 ∈ (1𝑜 +𝑜 1𝑜) |
10 | 1pi 9584 | . . . 4 ⊢ 1𝑜 ∈ N | |
11 | addpiord 9585 | . . . 4 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜)) | |
12 | 10, 10, 11 | mp2an 704 | . . 3 ⊢ (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜) |
13 | 9, 12 | eleqtrri 2687 | . 2 ⊢ 1𝑜 ∈ (1𝑜 +N 1𝑜) |
14 | addclpi 9593 | . . . 4 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) ∈ N) | |
15 | 10, 10, 14 | mp2an 704 | . . 3 ⊢ (1𝑜 +N 1𝑜) ∈ N |
16 | ltpiord 9588 | . . 3 ⊢ ((1𝑜 ∈ N ∧ (1𝑜 +N 1𝑜) ∈ N) → (1𝑜 <N (1𝑜 +N 1𝑜) ↔ 1𝑜 ∈ (1𝑜 +N 1𝑜))) | |
17 | 10, 15, 16 | mp2an 704 | . 2 ⊢ (1𝑜 <N (1𝑜 +N 1𝑜) ↔ 1𝑜 ∈ (1𝑜 +N 1𝑜)) |
18 | 13, 17 | mpbir 220 | 1 ⊢ 1𝑜 <N (1𝑜 +N 1𝑜) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∅c0 3874 class class class wbr 4583 (class class class)co 6549 ωcom 6957 1𝑜c1o 7440 +𝑜 coa 7444 Ncnpi 9545 +N cpli 9546 <N clti 9548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-ni 9573 df-pli 9574 df-lti 9576 |
This theorem is referenced by: 1lt2nq 9674 |
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