![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > onelssex | Structured version Visualization version GIF version |
Description: Ordinal less than is equivalent to having an ordinal between them. (Contributed by Scott Fenton, 8-Aug-2024.) |
Ref | Expression |
---|---|
onelssex | ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐶 ↔ ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 4018 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | sseq2 4022 | . . . 4 ⊢ (𝑏 = 𝐴 → (𝐴 ⊆ 𝑏 ↔ 𝐴 ⊆ 𝐴)) | |
3 | 2 | rspcev 3622 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐴) → ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏) |
4 | 1, 3 | mpan2 691 | . 2 ⊢ (𝐴 ∈ 𝐶 → ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏) |
5 | ontr2 6433 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝑏 ∧ 𝑏 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | |
6 | 5 | expcomd 416 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝑏 ∈ 𝐶 → (𝐴 ⊆ 𝑏 → 𝐴 ∈ 𝐶))) |
7 | 6 | rexlimdv 3151 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏 → 𝐴 ∈ 𝐶)) |
8 | 4, 7 | impbid2 226 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐶 ↔ ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 Oncon0 6386 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: madebdayim 27941 madebdaylemold 27951 |
Copyright terms: Public domain | W3C validator |