Theorem elwlim 36172

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 3020	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ↔ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅)))
2		id 22	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
3		predeq3 6293	. . . . . 6 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
4	3	supeq1d 9393	. . . . 5 ⊢ (𝑥 = 𝑋 → sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
5	2, 4	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) ↔ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))
6	1, 5	anbi12d 641	. . 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 36162	. . 3 ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
8	6, 7	elrab2 3655	. 2 ⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ (𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))))
9		3anass 1107	. 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 280	1 ⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  Predcpred 6288  supcsup 9387  infcinf 9388  WLimcwlim 36160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-xp 5654  df-cnv 5656  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-sup 9389  df-wlim 36162

This theorem is referenced by: (None)