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Mirrors > Home > MPE Home > Th. List > orddif | Structured version Visualization version GIF version |
Description: Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
Ref | Expression |
---|---|
orddif | ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orddisj 6197 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | |
2 | disj3 4361 | . . 3 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴})) | |
3 | df-suc 6165 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | difeq1i 4046 | . . . . 5 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴}) |
5 | difun2 4387 | . . . . 5 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴}) | |
6 | 4, 5 | eqtri 2821 | . . . 4 ⊢ (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴}) |
7 | 6 | eqeq2i 2811 | . . 3 ⊢ (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴})) |
8 | 2, 7 | bitr4i 281 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴})) |
9 | 1, 8 | sylib 221 | 1 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 {csn 4525 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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 df-fr 5478 df-we 5480 df-ord 6162 df-suc 6165 |
This theorem is referenced by: phplem3 8682 phplem4 8683 pssnn 8720 |
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