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Theorem cflem 9656
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 3986 . . 3 𝐴𝐴
2 ssid 3986 . . . . 5 𝑧𝑧
3 sseq2 3990 . . . . . 6 (𝑤 = 𝑧 → (𝑧𝑤𝑧𝑧))
43rspcev 3620 . . . . 5 ((𝑧𝐴𝑧𝑧) → ∃𝑤𝐴 𝑧𝑤)
52, 4mpan2 687 . . . 4 (𝑧𝐴 → ∃𝑤𝐴 𝑧𝑤)
65rgen 3145 . . 3 𝑧𝐴𝑤𝐴 𝑧𝑤
7 sseq1 3989 . . . . 5 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
8 rexeq 3404 . . . . . 6 (𝑦 = 𝐴 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝐴 𝑧𝑤))
98ralbidv 3194 . . . . 5 (𝑦 = 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
107, 9anbi12d 630 . . . 4 (𝑦 = 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)))
1110spcegv 3594 . . 3 (𝐴𝑉 → ((𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤) → ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
121, 6, 11mp2ani 694 . 2 (𝐴𝑉 → ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
13 fvex 6676 . . . . . 6 (card‘𝑦) ∈ V
1413isseti 3506 . . . . 5 𝑥 𝑥 = (card‘𝑦)
15 19.41v 1941 . . . . 5 (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (∃𝑥 𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1614, 15mpbiran 705 . . . 4 (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
1716exbii 1839 . . 3 (∃𝑦𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
18 excom 2159 . . 3 (∃𝑦𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1917, 18bitr3i 278 . 2 (∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
2012, 19sylib 219 1 (𝐴𝑉 → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  wral 3135  wrex 3136  wss 3933  cfv 6348  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-pr 4560  df-uni 4831  df-iota 6307  df-fv 6356
This theorem is referenced by:  cfval  9657  cff  9658  cff1  9668
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