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Mirrors > Home > ILE Home > Th. List > ltpiord | GIF version |
Description: Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
ltpiord | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lti 7115 | . . 3 ⊢ <N = ( E ∩ (N × N)) | |
2 | 1 | breqi 3935 | . 2 ⊢ (𝐴 <N 𝐵 ↔ 𝐴( E ∩ (N × N))𝐵) |
3 | brinxp 4607 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 E 𝐵 ↔ 𝐴( E ∩ (N × N))𝐵)) | |
4 | epelg 4212 | . . . 4 ⊢ (𝐵 ∈ N → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
5 | 4 | adantl 275 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
6 | 3, 5 | bitr3d 189 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴( E ∩ (N × N))𝐵 ↔ 𝐴 ∈ 𝐵)) |
7 | 2, 6 | syl5bb 191 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 ∩ cin 3070 class class class wbr 3929 E cep 4209 × cxp 4537 Ncnpi 7080 <N clti 7083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-eprel 4211 df-xp 4545 df-lti 7115 |
This theorem is referenced by: ltsopi 7128 pitric 7129 pitri3or 7130 ltdcpi 7131 ltexpi 7145 ltapig 7146 ltmpig 7147 1lt2pi 7148 nlt1pig 7149 archnqq 7225 prarloclemarch2 7227 prarloclemlt 7301 prarloclemn 7307 |
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