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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-el2oss1o | GIF version |
Description: Shorter proof of el2oss1o 6443 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-el2oss1o | ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 6423 | . . . 4 ⊢ 1o ∈ On | |
2 | 1 | ontrci 4427 | . . 3 ⊢ Tr 1o |
3 | trsucss 4423 | . . 3 ⊢ (Tr 1o → (𝐴 ∈ suc 1o → 𝐴 ⊆ 1o)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ suc 1o → 𝐴 ⊆ 1o) |
5 | df-2o 6417 | . 2 ⊢ 2o = suc 1o | |
6 | 4, 5 | eleq2s 2272 | 1 ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ⊆ wss 3129 Tr wtr 4101 suc csuc 4365 1oc1o 6409 2oc2o 6410 |
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-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 |
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-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-tr 4102 df-iord 4366 df-on 4368 df-suc 4371 df-1o 6416 df-2o 6417 |
This theorem is referenced by: (None) |
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