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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 6223 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | |
2 | disj3 4401 | . . 3 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴})) | |
3 | df-suc 6191 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | difeq1i 4094 | . . . . 5 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴}) |
5 | difun2 4427 | . . . . 5 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴}) | |
6 | 4, 5 | eqtri 2844 | . . . 4 ⊢ (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴}) |
7 | 6 | eqeq2i 2834 | . . 3 ⊢ (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴})) |
8 | 2, 7 | bitr4i 279 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴})) |
9 | 1, 8 | sylib 219 | 1 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∖ cdif 3932 ∪ cun 3933 ∩ cin 3934 ∅c0 4290 {csn 4559 Ord word 6184 suc csuc 6187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5059 df-opab 5121 df-eprel 5459 df-fr 5508 df-we 5510 df-ord 6188 df-suc 6191 |
This theorem is referenced by: phplem3 8687 phplem4 8688 pssnn 8725 |
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