![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > onsdominel | Structured version Visualization version GIF version |
Description: An ordinal with more elements of some type is larger. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
onsdominel | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐴 ∩ 𝐶) ≺ (𝐵 ∩ 𝐶)) → 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ontri1 6352 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵)) | |
2 | 1 | ancoms 460 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵)) |
3 | inex1g 5277 | . . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ∩ 𝐶) ∈ V) | |
4 | ssrin 4194 | . . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∩ 𝐶) ⊆ (𝐴 ∩ 𝐶)) | |
5 | ssdomg 8943 | . . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) ∈ V → ((𝐵 ∩ 𝐶) ⊆ (𝐴 ∩ 𝐶) → (𝐵 ∩ 𝐶) ≼ (𝐴 ∩ 𝐶))) | |
6 | 3, 4, 5 | syl2im 40 | . . . . . 6 ⊢ (𝐴 ∈ On → (𝐵 ⊆ 𝐴 → (𝐵 ∩ 𝐶) ≼ (𝐴 ∩ 𝐶))) |
7 | domnsym 9046 | . . . . . 6 ⊢ ((𝐵 ∩ 𝐶) ≼ (𝐴 ∩ 𝐶) → ¬ (𝐴 ∩ 𝐶) ≺ (𝐵 ∩ 𝐶)) | |
8 | 6, 7 | syl6 35 | . . . . 5 ⊢ (𝐴 ∈ On → (𝐵 ⊆ 𝐴 → ¬ (𝐴 ∩ 𝐶) ≺ (𝐵 ∩ 𝐶))) |
9 | 8 | adantr 482 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 → ¬ (𝐴 ∩ 𝐶) ≺ (𝐵 ∩ 𝐶))) |
10 | 2, 9 | sylbird 260 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ∈ 𝐵 → ¬ (𝐴 ∩ 𝐶) ≺ (𝐵 ∩ 𝐶))) |
11 | 10 | con4d 115 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∩ 𝐶) ≺ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵)) |
12 | 11 | 3impia 1118 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐴 ∩ 𝐶) ≺ (𝐵 ∩ 𝐶)) → 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 Vcvv 3444 ∩ cin 3910 ⊆ wss 3911 class class class wbr 5106 Oncon0 6318 ≼ cdom 8884 ≺ csdm 8885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 |
This theorem is referenced by: fin23lem27 10269 |
Copyright terms: Public domain | W3C validator |