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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 14623 | . . 3 ⊢ (𝐴 ∈ ω → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
2 | 1 | ralrimiv 2549 | . 2 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) |
3 | dftr3 4105 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) | |
4 | 2, 3 | sylibr 134 | 1 ⊢ (𝐴 ∈ ω → Tr 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 Tr wtr 4101 ωcom 4589 |
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-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-nul 4129 ax-pr 4209 ax-un 4433 ax-bd0 14485 ax-bdor 14488 ax-bdal 14490 ax-bdex 14491 ax-bdeq 14492 ax-bdel 14493 ax-bdsb 14494 ax-bdsep 14556 ax-infvn 14613 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-sn 3598 df-pr 3599 df-uni 3810 df-int 3845 df-tr 4102 df-suc 4371 df-iom 4590 df-bdc 14513 df-bj-ind 14599 |
This theorem is referenced by: bj-nnord 14630 bj-omord 14632 |
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