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| Mirrors > Home > MPE Home > Th. List > Mathboxes > onsucelab | Structured version Visualization version GIF version | ||
| Description: The successor of every ordinal is an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.) |
| Ref | Expression |
|---|---|
| onsucelab | ⊢ (𝐴 ∈ On → suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | onsuc 7831 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
| 2 | eqid 2737 | . . 3 ⊢ suc 𝐴 = suc 𝐴 | |
| 3 | id 22 | . . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ On) | |
| 4 | suceq 6450 | . . . . . 6 ⊢ (𝑏 = 𝐴 → suc 𝑏 = suc 𝐴) | |
| 5 | 4 | eqeq2d 2748 | . . . . 5 ⊢ (𝑏 = 𝐴 → (suc 𝐴 = suc 𝑏 ↔ suc 𝐴 = suc 𝐴)) |
| 6 | 5 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑏 = 𝐴) → (suc 𝐴 = suc 𝑏 ↔ suc 𝐴 = suc 𝐴)) |
| 7 | 3, 6 | rspcedv 3615 | . . 3 ⊢ (𝐴 ∈ On → (suc 𝐴 = suc 𝐴 → ∃𝑏 ∈ On suc 𝐴 = suc 𝑏)) |
| 8 | 2, 7 | mpi 20 | . 2 ⊢ (𝐴 ∈ On → ∃𝑏 ∈ On suc 𝐴 = suc 𝑏) |
| 9 | eqeq1 2741 | . . . 4 ⊢ (𝑎 = suc 𝐴 → (𝑎 = suc 𝑏 ↔ suc 𝐴 = suc 𝑏)) | |
| 10 | 9 | rexbidv 3179 | . . 3 ⊢ (𝑎 = suc 𝐴 → (∃𝑏 ∈ On 𝑎 = suc 𝑏 ↔ ∃𝑏 ∈ On suc 𝐴 = suc 𝑏)) |
| 11 | 10 | elrab 3692 | . 2 ⊢ (suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ↔ (suc 𝐴 ∈ On ∧ ∃𝑏 ∈ On suc 𝐴 = suc 𝑏)) |
| 12 | 1, 8, 11 | sylanbrc 583 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 Oncon0 6384 suc csuc 6386 |
| 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 ax-un 7755 |
| 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 df-suc 6390 |
| This theorem is referenced by: (None) |
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