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Mirrors > Home > MPE Home > Th. List > Mathboxes > onepsuc | Structured version Visualization version GIF version |
Description: Every ordinal is less than its successor, relationship version. Lemma 1.7 of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.) |
Ref | Expression |
---|---|
onepsuc | ⊢ (𝐴 ∈ On → 𝐴 E suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucidg 6463 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴) | |
2 | onsuc 7827 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
3 | epelg 5583 | . . 3 ⊢ (suc 𝐴 ∈ On → (𝐴 E suc 𝐴 ↔ 𝐴 ∈ suc 𝐴)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ On → (𝐴 E suc 𝐴 ↔ 𝐴 ∈ suc 𝐴)) |
5 | 1, 4 | mpbird 257 | 1 ⊢ (𝐴 ∈ On → 𝐴 E suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 class class class wbr 5141 E cep 5581 Oncon0 6382 suc csuc 6384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 ax-un 7751 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-tr 5258 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-ord 6385 df-on 6386 df-suc 6388 |
This theorem is referenced by: (None) |
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