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| Mirrors > Home > MPE Home > Th. List > Mathboxes > oneptr | Structured version Visualization version GIF version | ||
| Description: The strict order on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| oneptr | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | epweon 7796 | . 2 ⊢ E We On | |
| 2 | weso 5675 | . 2 ⊢ ( E We On → E Or On) | |
| 3 | sopo 5610 | . . 3 ⊢ ( E Or On → E Po On) | |
| 4 | potr 5604 | . . . 4 ⊢ (( E Po On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶)) | |
| 5 | 4 | ex 412 | . . 3 ⊢ ( E Po On → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶))) | 
| 6 | 3, 5 | syl 17 | . 2 ⊢ ( E Or On → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶))) | 
| 7 | 1, 2, 6 | mp2b 10 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2107 class class class wbr 5142 E cep 5582 Po wpo 5589 Or wor 5590 We wwe 5635 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 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 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: (None) | 
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