Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sdomel | Structured version Visualization version GIF version |
Description: Strict dominance implies ordinal membership. (Contributed by Mario Carneiro, 13-Jan-2013.) |
Ref | Expression |
---|---|
sdomel | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≺ 𝐵 → 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdomg 8549 | . . . . 5 ⊢ (𝐴 ∈ On → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
2 | 1 | adantl 484 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) |
3 | ontri1 6219 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵)) | |
4 | domtriord 8657 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝐵)) | |
5 | 2, 3, 4 | 3imtr3d 295 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ≺ 𝐵)) |
6 | 5 | con4d 115 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ≺ 𝐵 → 𝐴 ∈ 𝐵)) |
7 | 6 | ancoms 461 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≺ 𝐵 → 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5058 Oncon0 6185 ≼ cdom 8501 ≺ csdm 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 |
This theorem is referenced by: findcard3 8755 harval2 9420 alephsuc2 9500 inawinalem 10105 |
Copyright terms: Public domain | W3C validator |