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Theorem cflem 8951
Description: A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set 𝐴. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
cflem (𝐴𝑉 → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝐴
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cflem
StepHypRef Expression
1 ssid 3587 . . 3 𝐴𝐴
2 ssid 3587 . . . . 5 𝑧𝑧
3 sseq2 3590 . . . . . 6 (𝑤 = 𝑧 → (𝑧𝑤𝑧𝑧))
43rspcev 3282 . . . . 5 ((𝑧𝐴𝑧𝑧) → ∃𝑤𝐴 𝑧𝑤)
52, 4mpan2 703 . . . 4 (𝑧𝐴 → ∃𝑤𝐴 𝑧𝑤)
65rgen 2906 . . 3 𝑧𝐴𝑤𝐴 𝑧𝑤
7 sseq1 3589 . . . . 5 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
8 rexeq 3116 . . . . . 6 (𝑦 = 𝐴 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝐴 𝑧𝑤))
98ralbidv 2969 . . . . 5 (𝑦 = 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
107, 9anbi12d 743 . . . 4 (𝑦 = 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)))
1110spcegv 3267 . . 3 (𝐴𝑉 → ((𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤) → ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
121, 6, 11mp2ani 710 . 2 (𝐴𝑉 → ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
13 fvex 6113 . . . . . 6 (card‘𝑦) ∈ V
1413isseti 3182 . . . . 5 𝑥 𝑥 = (card‘𝑦)
15 19.41v 1901 . . . . 5 (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (∃𝑥 𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1614, 15mpbiran 955 . . . 4 (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
1716exbii 1764 . . 3 (∃𝑦𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
18 excom 2029 . . 3 (∃𝑦𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1917, 18bitr3i 265 . 2 (∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
2012, 19sylib 207 1 (𝐴𝑉 → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wex 1695  wcel 1977  wral 2896  wrex 2897  wss 3540  cfv 5804  cardccrd 8644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768  df-fv 5812
This theorem is referenced by:  cfval  8952  cff  8953  cff1  8963
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