Theorem elwlim 36483

Description: Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)

Assertion

Ref	Expression
elwlim	⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))

Proof of Theorem elwlim

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 3017	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ↔ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅)))
2		id 23	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
3		predeq3 6305	. . . . . 6 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
4	3	supeq1d 9423	. . . . 5 ⊢ (𝑥 = 𝑋 → sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
5	2, 4	eqeq12d 2776	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) ↔ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))
6	1, 5	anbi12d 644	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅)) ↔ (𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))))
7		df-wlim 36473	. . 3 ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
8	6, 7	elrab2 3649	. 2 ⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ (𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))))
9		3anass 1111	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)) ↔ (𝑋 ∈ 𝐴 ∧ (𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))))
10	8, 9	bitr4i 281	1 ⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  Predcpred 6300  supcsup 9417  infcinf 9418  WLimcwlim 36471

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-sup 9419  df-wlim 36473

This theorem is used by: (None)