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Mirrors > Home > MPE Home > Th. List > ltsopi | Structured version Visualization version GIF version |
Description: Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltsopi | ⊢ <N Or N |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ni 9573 | . . . 4 ⊢ N = (ω ∖ {∅}) | |
2 | difss 3699 | . . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω | |
3 | omsson 6961 | . . . . 5 ⊢ ω ⊆ On | |
4 | 2, 3 | sstri 3577 | . . . 4 ⊢ (ω ∖ {∅}) ⊆ On |
5 | 1, 4 | eqsstri 3598 | . . 3 ⊢ N ⊆ On |
6 | epweon 6875 | . . . 4 ⊢ E We On | |
7 | weso 5029 | . . . 4 ⊢ ( E We On → E Or On) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ E Or On |
9 | soss 4977 | . . 3 ⊢ (N ⊆ On → ( E Or On → E Or N)) | |
10 | 5, 8, 9 | mp2 9 | . 2 ⊢ E Or N |
11 | df-lti 9576 | . . . 4 ⊢ <N = ( E ∩ (N × N)) | |
12 | soeq1 4978 | . . . 4 ⊢ ( <N = ( E ∩ (N × N)) → ( <N Or N ↔ ( E ∩ (N × N)) Or N)) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ ( <N Or N ↔ ( E ∩ (N × N)) Or N) |
14 | soinxp 5106 | . . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N) | |
15 | 13, 14 | bitr4i 266 | . 2 ⊢ ( <N Or N ↔ E Or N) |
16 | 10, 15 | mpbir 220 | 1 ⊢ <N Or N |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 E cep 4947 Or wor 4958 We wwe 4996 × cxp 5036 Oncon0 5640 ωcom 6957 Ncnpi 9545 <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-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-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 df-ni 9573 df-lti 9576 |
This theorem is referenced by: indpi 9608 nqereu 9630 ltsonq 9670 archnq 9681 |
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