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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 6467 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴) | |
2 | eloni 6396 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | ordsuc 7833 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
4 | 2, 3 | sylib 218 | . . 3 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
5 | ordelpss 6414 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) | |
6 | 2, 4, 5 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) |
7 | 1, 6 | mpbid 232 | 1 ⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 ⊊ wpss 3964 Ord word 6385 Oncon0 6386 suc csuc 6388 |
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 df-suc 6392 |
This theorem is referenced by: ackbij1lem15 10271 |
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