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Theorem iscard2 9251
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 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem iscard2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardon 9219 . . 3 (card‘𝐴) ∈ On
2 eleq1 2870 . . 3 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 234 . 2 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 cardonle 9232 . . . . . 6 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
54biantrurd 533 . . . . 5 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴))))
6 eqss 3904 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
75, 6syl6rbbr 291 . . . 4 (𝐴 ∈ On → ((card‘𝐴) = 𝐴𝐴 ⊆ (card‘𝐴)))
8 oncardval 9230 . . . . 5 (𝐴 ∈ On → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
98sseq2d 3920 . . . 4 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ 𝐴 {𝑦 ∈ On ∣ 𝑦𝐴}))
107, 9bitrd 280 . . 3 (𝐴 ∈ On → ((card‘𝐴) = 𝐴𝐴 {𝑦 ∈ On ∣ 𝑦𝐴}))
11 ssint 4798 . . . 4 (𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴}𝐴𝑥)
12 breq1 4965 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
1312elrab 3618 . . . . . . . 8 (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
14 ensymb 8405 . . . . . . . . 9 (𝑥𝐴𝐴𝑥)
1514anbi2i 622 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (𝑥 ∈ On ∧ 𝐴𝑥))
1613, 15bitri 276 . . . . . . 7 (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ (𝑥 ∈ On ∧ 𝐴𝑥))
1716imbi1i 351 . . . . . 6 ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} → 𝐴𝑥) ↔ ((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴𝑥))
18 impexp 451 . . . . . 6 (((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴𝑥) ↔ (𝑥 ∈ On → (𝐴𝑥𝐴𝑥)))
1917, 18bitri 276 . . . . 5 ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} → 𝐴𝑥) ↔ (𝑥 ∈ On → (𝐴𝑥𝐴𝑥)))
2019ralbii2 3130 . . . 4 (∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴}𝐴𝑥 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥))
2111, 20bitri 276 . . 3 (𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥))
2210, 21syl6bb 288 . 2 (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
233, 22biadanii 819 1 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  {crab 3109  wss 3859   cint 4782   class class class wbr 4962  Oncon0 6066  cfv 6225  cen 8354  cardccrd 9210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-er 8139  df-en 8358  df-card 9214
This theorem is referenced by:  harcard  9253
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