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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-omtrans2 | GIF version |
Description: The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-omtrans2 | ⊢ Tr ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr3 4078 | . 2 ⊢ (Tr ω ↔ ∀𝑥 ∈ ω 𝑥 ⊆ ω) | |
2 | bj-omtrans 13673 | . 2 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ ω) | |
3 | 1, 2 | mprgbir 2522 | 1 ⊢ Tr ω |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3111 Tr wtr 4074 ωcom 4561 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-nul 4102 ax-pr 4181 ax-un 4405 ax-bd0 13530 ax-bdor 13533 ax-bdal 13535 ax-bdex 13536 ax-bdeq 13537 ax-bdel 13538 ax-bdsb 13539 ax-bdsep 13601 ax-infvn 13658 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-tr 4075 df-suc 4343 df-iom 4562 df-bdc 13558 df-bj-ind 13644 |
This theorem is referenced by: bj-omord 13677 |
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