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Mirrors > Home > MPE Home > Th. List > sucidg | Structured version Visualization version GIF version |
Description: Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.) |
Ref | Expression |
---|---|
sucidg | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | olci 855 | . 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴) |
3 | elsucg 5933 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 248 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 836 = wceq 1631 ∈ wcel 2145 suc csuc 5866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-v 3353 df-un 3728 df-sn 4317 df-suc 5870 |
This theorem is referenced by: sucid 5945 nsuceq0 5946 trsuc 5951 sucssel 5960 ordsuc 7159 onpsssuc 7164 nlimsucg 7187 tfrlem11 7635 tfrlem13 7637 tz7.44-2 7654 omeulem1 7814 oeordi 7819 oeeulem 7833 php4 8301 wofib 8604 suc11reg 8678 cantnfle 8730 cantnflt2 8732 cantnfp1lem3 8739 cantnflem1 8748 dfac12lem1 9165 dfac12lem2 9166 ttukeylem3 9533 ttukeylem7 9537 r1wunlim 9759 noresle 32176 noprefixmo 32178 ontgval 32760 sucneqond 33543 finxpreclem4 33561 finxpsuclem 33564 |
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