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Theorem pm54.43lem 9941
Description: In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 9909), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = 1o}. Here we show that this is equivalent to 𝐴 β‰ˆ 1o so that we can use the latter more convenient notation in pm54.43 9942. (Contributed by NM, 4-Nov-2013.)
Assertion
Ref Expression
pm54.43lem (𝐴 β‰ˆ 1o ↔ 𝐴 ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = 1o})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem pm54.43lem
StepHypRef Expression
1 carden2b 9908 . . . 4 (𝐴 β‰ˆ 1o β†’ (cardβ€˜π΄) = (cardβ€˜1o))
2 1onn 8587 . . . . 5 1o ∈ Ο‰
3 cardnn 9904 . . . . 5 (1o ∈ Ο‰ β†’ (cardβ€˜1o) = 1o)
42, 3ax-mp 5 . . . 4 (cardβ€˜1o) = 1o
51, 4eqtrdi 2789 . . 3 (𝐴 β‰ˆ 1o β†’ (cardβ€˜π΄) = 1o)
64eqeq2i 2746 . . . . 5 ((cardβ€˜π΄) = (cardβ€˜1o) ↔ (cardβ€˜π΄) = 1o)
76biimpri 227 . . . 4 ((cardβ€˜π΄) = 1o β†’ (cardβ€˜π΄) = (cardβ€˜1o))
8 1n0 8435 . . . . . . . 8 1o β‰  βˆ…
98neii 2942 . . . . . . 7 Β¬ 1o = βˆ…
10 eqeq1 2737 . . . . . . 7 ((cardβ€˜π΄) = 1o β†’ ((cardβ€˜π΄) = βˆ… ↔ 1o = βˆ…))
119, 10mtbiri 327 . . . . . 6 ((cardβ€˜π΄) = 1o β†’ Β¬ (cardβ€˜π΄) = βˆ…)
12 ndmfv 6878 . . . . . 6 (Β¬ 𝐴 ∈ dom card β†’ (cardβ€˜π΄) = βˆ…)
1311, 12nsyl2 141 . . . . 5 ((cardβ€˜π΄) = 1o β†’ 𝐴 ∈ dom card)
14 1on 8425 . . . . . 6 1o ∈ On
15 onenon 9890 . . . . . 6 (1o ∈ On β†’ 1o ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1o ∈ dom card
17 carden2 9928 . . . . 5 ((𝐴 ∈ dom card ∧ 1o ∈ dom card) β†’ ((cardβ€˜π΄) = (cardβ€˜1o) ↔ 𝐴 β‰ˆ 1o))
1813, 16, 17sylancl 587 . . . 4 ((cardβ€˜π΄) = 1o β†’ ((cardβ€˜π΄) = (cardβ€˜1o) ↔ 𝐴 β‰ˆ 1o))
197, 18mpbid 231 . . 3 ((cardβ€˜π΄) = 1o β†’ 𝐴 β‰ˆ 1o)
205, 19impbii 208 . 2 (𝐴 β‰ˆ 1o ↔ (cardβ€˜π΄) = 1o)
21 fveqeq2 6852 . . 3 (π‘₯ = 𝐴 β†’ ((cardβ€˜π‘₯) = 1o ↔ (cardβ€˜π΄) = 1o))
2213, 21elab3 3639 . 2 (𝐴 ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = 1o} ↔ (cardβ€˜π΄) = 1o)
2320, 22bitr4i 278 1 (𝐴 β‰ˆ 1o ↔ 𝐴 ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = 1o})
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ…c0 4283   class class class wbr 5106  dom cdm 5634  Oncon0 6318  β€˜cfv 6497  Ο‰com 7803  1oc1o 8406   β‰ˆ cen 8883  cardccrd 9876
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9880
This theorem is referenced by: (None)
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