Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ordtr2 | Structured version Visualization version GIF version |
Description: Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
ordtr2 | ⊢ ((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordelord 6206 | . . . . . . . 8 ⊢ ((Ord 𝐶 ∧ 𝐵 ∈ 𝐶) → Ord 𝐵) | |
2 | 1 | ex 413 | . . . . . . 7 ⊢ (Ord 𝐶 → (𝐵 ∈ 𝐶 → Ord 𝐵)) |
3 | 2 | ancld 551 | . . . . . 6 ⊢ (Ord 𝐶 → (𝐵 ∈ 𝐶 → (𝐵 ∈ 𝐶 ∧ Ord 𝐵))) |
4 | 3 | anc2li 556 | . . . . 5 ⊢ (Ord 𝐶 → (𝐵 ∈ 𝐶 → (Ord 𝐶 ∧ (𝐵 ∈ 𝐶 ∧ Ord 𝐵)))) |
5 | ordelpss 6212 | . . . . . . . . . 10 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ∈ 𝐶 ↔ 𝐵 ⊊ 𝐶)) | |
6 | sspsstr 4079 | . . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
7 | 6 | expcom 414 | . . . . . . . . . 10 ⊢ (𝐵 ⊊ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊊ 𝐶)) |
8 | 5, 7 | syl6bi 254 | . . . . . . . . 9 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊊ 𝐶))) |
9 | 8 | expcom 414 | . . . . . . . 8 ⊢ (Ord 𝐶 → (Ord 𝐵 → (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊊ 𝐶)))) |
10 | 9 | com23 86 | . . . . . . 7 ⊢ (Ord 𝐶 → (𝐵 ∈ 𝐶 → (Ord 𝐵 → (𝐴 ⊆ 𝐵 → 𝐴 ⊊ 𝐶)))) |
11 | 10 | imp32 419 | . . . . . 6 ⊢ ((Ord 𝐶 ∧ (𝐵 ∈ 𝐶 ∧ Ord 𝐵)) → (𝐴 ⊆ 𝐵 → 𝐴 ⊊ 𝐶)) |
12 | 11 | com12 32 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ((Ord 𝐶 ∧ (𝐵 ∈ 𝐶 ∧ Ord 𝐵)) → 𝐴 ⊊ 𝐶)) |
13 | 4, 12 | syl9 77 | . . . 4 ⊢ (Ord 𝐶 → (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐶 → 𝐴 ⊊ 𝐶))) |
14 | 13 | impd 411 | . . 3 ⊢ (Ord 𝐶 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊊ 𝐶)) |
15 | 14 | adantl 482 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊊ 𝐶)) |
16 | ordelpss 6212 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐶) → (𝐴 ∈ 𝐶 ↔ 𝐴 ⊊ 𝐶)) | |
17 | 15, 16 | sylibrd 260 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 ⊊ wpss 3934 Ord word 6183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 |
This theorem is referenced by: ontr2 6231 ordelinel 6282 smogt 7993 smorndom 7994 nnarcl 8231 nnawordex 8252 coftr 9683 noetalem3 33116 hfuni 33542 |
Copyright terms: Public domain | W3C validator |