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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 7337 | . . 3 ⊢ <N = ( E ∩ (N × N)) | |
2 | 1 | breqi 4024 | . 2 ⊢ (𝐴 <N 𝐵 ↔ 𝐴( E ∩ (N × N))𝐵) |
3 | brinxp 4712 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 E 𝐵 ↔ 𝐴( E ∩ (N × N))𝐵)) | |
4 | epelg 4308 | . . . 4 ⊢ (𝐵 ∈ N → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
5 | 4 | adantl 277 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
6 | 3, 5 | bitr3d 190 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴( E ∩ (N × N))𝐵 ↔ 𝐴 ∈ 𝐵)) |
7 | 2, 6 | bitrid 192 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2160 ∩ cin 3143 class class class wbr 4018 E cep 4305 × cxp 4642 Ncnpi 7302 <N clti 7305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-eprel 4307 df-xp 4650 df-lti 7337 |
This theorem is referenced by: ltsopi 7350 pitric 7351 pitri3or 7352 ltdcpi 7353 ltexpi 7367 ltapig 7368 ltmpig 7369 1lt2pi 7370 nlt1pig 7371 archnqq 7447 prarloclemarch2 7449 prarloclemlt 7523 prarloclemn 7529 |
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