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Theorem iscard2 9917
Description: Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
iscard2 ((cardβ€˜π΄) = 𝐴 ↔ (𝐴 ∈ On ∧ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
Distinct variable group:   π‘₯,𝐴

Proof of Theorem iscard2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardon 9885 . . 3 (cardβ€˜π΄) ∈ On
2 eleq1 2822 . . 3 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 232 . 2 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
4 eqss 3960 . . . . 5 ((cardβ€˜π΄) = 𝐴 ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄)))
5 cardonle 9898 . . . . . 6 (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
65biantrurd 534 . . . . 5 (𝐴 ∈ On β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄))))
74, 6bitr4id 290 . . . 4 (𝐴 ∈ On β†’ ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 βŠ† (cardβ€˜π΄)))
8 oncardval 9896 . . . . 5 (𝐴 ∈ On β†’ (cardβ€˜π΄) = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
98sseq2d 3977 . . . 4 (𝐴 ∈ On β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ 𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}))
107, 9bitrd 279 . . 3 (𝐴 ∈ On β†’ ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}))
11 ssint 4926 . . . 4 (𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ βˆ€π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}𝐴 βŠ† π‘₯)
12 breq1 5109 . . . . . . . . 9 (𝑦 = π‘₯ β†’ (𝑦 β‰ˆ 𝐴 ↔ π‘₯ β‰ˆ 𝐴))
1312elrab 3646 . . . . . . . 8 (π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ (π‘₯ ∈ On ∧ π‘₯ β‰ˆ 𝐴))
14 ensymb 8945 . . . . . . . . 9 (π‘₯ β‰ˆ 𝐴 ↔ 𝐴 β‰ˆ π‘₯)
1514anbi2i 624 . . . . . . . 8 ((π‘₯ ∈ On ∧ π‘₯ β‰ˆ 𝐴) ↔ (π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯))
1613, 15bitri 275 . . . . . . 7 (π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ (π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯))
1716imbi1i 350 . . . . . 6 ((π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} β†’ 𝐴 βŠ† π‘₯) ↔ ((π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯) β†’ 𝐴 βŠ† π‘₯))
18 impexp 452 . . . . . 6 (((π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯) β†’ 𝐴 βŠ† π‘₯) ↔ (π‘₯ ∈ On β†’ (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
1917, 18bitri 275 . . . . 5 ((π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} β†’ 𝐴 βŠ† π‘₯) ↔ (π‘₯ ∈ On β†’ (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
2019ralbii2 3089 . . . 4 (βˆ€π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}𝐴 βŠ† π‘₯ ↔ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯))
2111, 20bitri 275 . . 3 (𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯))
2210, 21bitrdi 287 . 2 (𝐴 ∈ On β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
233, 22biadanii 821 1 ((cardβ€˜π΄) = 𝐴 ↔ (𝐴 ∈ On ∧ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406   βŠ† wss 3911  βˆ© cint 4908   class class class wbr 5106  Oncon0 6318  β€˜cfv 6497   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-rab 3407  df-v 3446  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-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8651  df-en 8887  df-card 9880
This theorem is referenced by:  harcard  9919
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