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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 8661 | . . . . 5 ⊢ 1o ∈ ω | |
2 | nna0 8625 | . . . . 5 ⊢ (1o ∈ ω → (1o +o ∅) = 1o) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1o +o ∅) = 1o |
4 | 0lt1o 8525 | . . . . 5 ⊢ ∅ ∈ 1o | |
5 | peano1 7895 | . . . . . 6 ⊢ ∅ ∈ ω | |
6 | nnaord 8640 | . . . . . 6 ⊢ ((∅ ∈ ω ∧ 1o ∈ ω ∧ 1o ∈ ω) → (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o))) | |
7 | 5, 1, 1, 6 | mp3an 1457 | . . . . 5 ⊢ (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o)) |
8 | 4, 7 | mpbi 229 | . . . 4 ⊢ (1o +o ∅) ∈ (1o +o 1o) |
9 | 3, 8 | eqeltrri 2822 | . . 3 ⊢ 1o ∈ (1o +o 1o) |
10 | 1pi 10908 | . . . 4 ⊢ 1o ∈ N | |
11 | addpiord 10909 | . . . 4 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) = (1o +o 1o)) | |
12 | 10, 10, 11 | mp2an 690 | . . 3 ⊢ (1o +N 1o) = (1o +o 1o) |
13 | 9, 12 | eleqtrri 2824 | . 2 ⊢ 1o ∈ (1o +N 1o) |
14 | addclpi 10917 | . . . 4 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N) | |
15 | 10, 10, 14 | mp2an 690 | . . 3 ⊢ (1o +N 1o) ∈ N |
16 | ltpiord 10912 | . . 3 ⊢ ((1o ∈ N ∧ (1o +N 1o) ∈ N) → (1o <N (1o +N 1o) ↔ 1o ∈ (1o +N 1o))) | |
17 | 10, 15, 16 | mp2an 690 | . 2 ⊢ (1o <N (1o +N 1o) ↔ 1o ∈ (1o +N 1o)) |
18 | 13, 17 | mpbir 230 | 1 ⊢ 1o <N (1o +N 1o) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∅c0 4322 class class class wbr 5149 (class class class)co 7419 ωcom 7871 1oc1o 8480 +o coa 8484 Ncnpi 10869 +N cpli 10870 <N clti 10872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-ni 10897 df-pli 10898 df-lti 10900 |
This theorem is referenced by: 1lt2nq 10998 |
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