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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-el2oss1o | GIF version |
Description: Shorter proof of el2oss1o 6402 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 6382 | . . . 4 ⊢ 1o ∈ On | |
2 | 1 | ontrci 4399 | . . 3 ⊢ Tr 1o |
3 | trsucss 4395 | . . 3 ⊢ (Tr 1o → (𝐴 ∈ suc 1o → 𝐴 ⊆ 1o)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ suc 1o → 𝐴 ⊆ 1o) |
5 | df-2o 6376 | . 2 ⊢ 2o = suc 1o | |
6 | 4, 5 | eleq2s 2259 | 1 ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 ⊆ wss 3111 Tr wtr 4074 suc csuc 4337 1oc1o 6368 2oc2o 6369 |
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-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
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-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-tr 4075 df-iord 4338 df-on 4340 df-suc 4343 df-1o 6375 df-2o 6376 |
This theorem is referenced by: (None) |
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