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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nntrans2 | GIF version |
Description: A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nntrans2 | ⊢ (𝐴 ∈ ω → Tr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nntrans 13568 | . . 3 ⊢ (𝐴 ∈ ω → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
2 | 1 | ralrimiv 2529 | . 2 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) |
3 | dftr3 4067 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) | |
4 | 2, 3 | sylibr 133 | 1 ⊢ (𝐴 ∈ ω → Tr 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 ∀wral 2435 ⊆ wss 3102 Tr wtr 4063 ωcom 4550 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-nul 4091 ax-pr 4170 ax-un 4394 ax-bd0 13430 ax-bdor 13433 ax-bdal 13435 ax-bdex 13436 ax-bdeq 13437 ax-bdel 13438 ax-bdsb 13439 ax-bdsep 13501 ax-infvn 13558 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-sn 3566 df-pr 3567 df-uni 3774 df-int 3809 df-tr 4064 df-suc 4332 df-iom 4551 df-bdc 13458 df-bj-ind 13544 |
This theorem is referenced by: bj-nnord 13575 bj-omord 13577 |
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