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| Mirrors > Home > MPE Home > Th. List > Mathboxes > onexgt | Structured version Visualization version GIF version | ||
| Description: For any ordinal, there is always a larger ordinal. (Contributed by RP, 1-Feb-2025.) |
| Ref | Expression |
|---|---|
| onexgt | ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | onsuc 7831 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
| 2 | sucidg 6465 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴) | |
| 3 | eleq2 2830 | . . 3 ⊢ (𝑥 = suc 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝐴)) | |
| 4 | 3 | rspcev 3622 | . 2 ⊢ ((suc 𝐴 ∈ On ∧ 𝐴 ∈ suc 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ 𝑥) |
| 5 | 1, 2, 4 | syl2anc 584 | 1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∃wrex 3070 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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