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| 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 4006 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
| 2 | sseq2 4010 | . . . 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 6431 | . . . 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 3153 | . 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 2108 ∃wrex 3070 ⊆ wss 3951 Oncon0 6384 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 | 
| This theorem is referenced by: madebdayim 27926 madebdaylemold 27936 | 
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