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Mirrors > Home > MPE Home > Th. List > onpsssuc | Structured version Visualization version GIF version |
Description: An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
Ref | Expression |
---|---|
onpsssuc | ⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucidg 6444 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴) | |
2 | eloni 6373 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | ordsuc 7810 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
4 | 2, 3 | sylib 217 | . . 3 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
5 | ordelpss 6391 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) | |
6 | 2, 4, 5 | syl2anc 583 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) |
7 | 1, 6 | mpbid 231 | 1 ⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2099 ⊊ wpss 3945 Ord word 6362 Oncon0 6363 suc csuc 6365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6366 df-on 6367 df-suc 6369 |
This theorem is referenced by: ackbij1lem15 10249 |
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