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Theorem cflem 10241
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 (𝐴 ∈ 𝑉 β†’ βˆƒπ‘₯βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
Distinct variable group:   π‘₯,𝑦,𝑧,𝑀,𝐴
Allowed substitution hints:   𝑉(π‘₯,𝑦,𝑧,𝑀)

Proof of Theorem cflem
StepHypRef Expression
1 ssid 4005 . . 3 𝐴 βŠ† 𝐴
2 ssid 4005 . . . . 5 𝑧 βŠ† 𝑧
3 sseq2 4009 . . . . . 6 (𝑀 = 𝑧 β†’ (𝑧 βŠ† 𝑀 ↔ 𝑧 βŠ† 𝑧))
43rspcev 3613 . . . . 5 ((𝑧 ∈ 𝐴 ∧ 𝑧 βŠ† 𝑧) β†’ βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)
52, 4mpan2 690 . . . 4 (𝑧 ∈ 𝐴 β†’ βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)
65rgen 3064 . . 3 βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀
7 sseq1 4008 . . . . 5 (𝑦 = 𝐴 β†’ (𝑦 βŠ† 𝐴 ↔ 𝐴 βŠ† 𝐴))
8 rexeq 3322 . . . . . 6 (𝑦 = 𝐴 β†’ (βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀))
98ralbidv 3178 . . . . 5 (𝑦 = 𝐴 β†’ (βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀))
107, 9anbi12d 632 . . . 4 (𝑦 = 𝐴 β†’ ((𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀) ↔ (𝐴 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)))
1110spcegv 3588 . . 3 (𝐴 ∈ 𝑉 β†’ ((𝐴 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀) β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
121, 6, 11mp2ani 697 . 2 (𝐴 ∈ 𝑉 β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
13 fvex 6905 . . . . . 6 (cardβ€˜π‘¦) ∈ V
1413isseti 3490 . . . . 5 βˆƒπ‘₯ π‘₯ = (cardβ€˜π‘¦)
15 19.41v 1954 . . . . 5 (βˆƒπ‘₯(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ (βˆƒπ‘₯ π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
1614, 15mpbiran 708 . . . 4 (βˆƒπ‘₯(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
1716exbii 1851 . . 3 (βˆƒπ‘¦βˆƒπ‘₯(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ βˆƒπ‘¦(𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
18 excom 2163 . . 3 (βˆƒπ‘¦βˆƒπ‘₯(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ βˆƒπ‘₯βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
1917, 18bitr3i 277 . 2 (βˆƒπ‘¦(𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀) ↔ βˆƒπ‘₯βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
2012, 19sylib 217 1 (𝐴 ∈ 𝑉 β†’ βˆƒπ‘₯βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   βŠ† wss 3949  β€˜cfv 6544  cardccrd 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-pr 4632  df-uni 4910  df-iota 6496  df-fv 6552
This theorem is referenced by:  cfval  10242  cff  10243  cff1  10253
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