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Mirrors > Home > MPE Home > Th. List > 0elsuc | Structured version Visualization version GIF version |
Description: The successor of an ordinal class contains the empty set. (Contributed by NM, 4-Apr-1995.) |
Ref | Expression |
---|---|
0elsuc | ⊢ (Ord 𝐴 → ∅ ∈ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsuc 7612 | . 2 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
2 | nsuceq0 6311 | . . 3 ⊢ suc 𝐴 ≠ ∅ | |
3 | ord0eln0 6285 | . . 3 ⊢ (Ord suc 𝐴 → (∅ ∈ suc 𝐴 ↔ suc 𝐴 ≠ ∅)) | |
4 | 2, 3 | mpbiri 261 | . 2 ⊢ (Ord suc 𝐴 → ∅ ∈ suc 𝐴) |
5 | 1, 4 | sylbi 220 | 1 ⊢ (Ord 𝐴 → ∅ ∈ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ≠ wne 2941 ∅c0 4252 Ord word 6230 suc csuc 6233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-11 2159 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-tr 5177 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-ord 6234 df-on 6235 df-suc 6237 |
This theorem is referenced by: oesuclem 8273 axdc3lem2 10090 axdc3lem4 10092 ssttrcl 33540 ttrcltr 33541 ttrclss 33545 ttrclselem2 33551 |
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