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Theorem elwlim 32095
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) ↔ (𝑋𝐴𝑋 ≠ 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.
StepHypRef Expression
1 neeq1 2994 . . . 4 (𝑥 = 𝑋 → (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ↔ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅)))
2 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
3 predeq3 5845 . . . . . 6 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
43supeq1d 8519 . . . . 5 (𝑥 = 𝑋 → sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
52, 4eqeq12d 2775 . . . 4 (𝑥 = 𝑋 → (𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) ↔ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))
61, 5anbi12d 749 . . 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 32085 . . 3 WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
86, 7elrab2 3507 . 2 (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋𝐴 ∧ (𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))))
9 3anass 1081 . 2 ((𝑋𝐴𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)) ↔ (𝑋𝐴 ∧ (𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))))
108, 9bitr4i 267 1 (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋𝐴𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  Predcpred 5840  supcsup 8513  infcinf 8514  WLimcwlim 32083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-sup 8515  df-wlim 32085
This theorem is referenced by: (None)
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