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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 7509 | . 2 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
2 | nsuceq0 6239 | . . 3 ⊢ suc 𝐴 ≠ ∅ | |
3 | ord0eln0 6213 | . . 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 2987 ∅c0 4243 Ord word 6158 suc csuc 6161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 df-suc 6165 |
This theorem is referenced by: oesuclem 8133 axdc3lem2 9862 axdc3lem4 9864 |
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