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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). Lemma 1.7 of [Schloeder] p. 1. (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 2735 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | olci 866 | . 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴) |
3 | elsucg 6454 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 258 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 847 = wceq 1537 ∈ wcel 2106 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 df-suc 6392 |
This theorem is referenced by: sucid 6468 nsuceq0 6469 trsuc 6473 sucssel 6481 ordsuc 7833 ordsucOLD 7834 onpsssuc 7839 nlimsucg 7863 tfrlem11 8427 tfrlem13 8429 tz7.44-2 8446 omeulem1 8619 oeordi 8624 oeeulem 8638 dif1enlem 9195 dif1enlemOLD 9196 rexdif1en 9197 rexdif1enOLD 9198 dif1en 9199 dif1enOLD 9201 php4 9248 wofib 9583 suc11reg 9657 cantnfle 9709 cantnflt2 9711 cantnfp1lem3 9718 cantnflem1 9727 dfac12lem1 10182 dfac12lem2 10183 ttukeylem3 10549 ttukeylem7 10553 r1wunlim 10775 noresle 27757 nosupprefixmo 27760 noinfprefixmo 27761 fmla 35366 ex-sategoelelomsuc 35411 ontgval 36414 sucneqond 37348 finxpreclem4 37377 finxpsuclem 37380 onexgt 43229 onepsuc 43241 ordnexbtwnsuc 43257 nlimsuc 43431 sucomisnotcard 43534 |
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