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Theorem card2on 9549
Description: The alternate definition of the cardinal of a set given in cardval2 9986 always gives a set, and indeed an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)
Assertion
Ref Expression
card2on {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On
Distinct variable group:   𝑥,𝐴

Proof of Theorem card2on
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onelon 6390 . . . . . . . . . . . . 13 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
2 vex 3479 . . . . . . . . . . . . . 14 𝑧 ∈ V
3 onelss 6407 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
43imp 408 . . . . . . . . . . . . . 14 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
5 ssdomg 8996 . . . . . . . . . . . . . 14 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
62, 4, 5mpsyl 68 . . . . . . . . . . . . 13 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
71, 6jca 513 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → (𝑦 ∈ On ∧ 𝑦𝑧))
8 domsdomtr 9112 . . . . . . . . . . . . . 14 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
98anim2i 618 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ (𝑦𝑧𝑧𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
109anassrs 469 . . . . . . . . . . . 12 (((𝑦 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
117, 10sylan 581 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
1211exp31 421 . . . . . . . . . 10 (𝑧 ∈ On → (𝑦𝑧 → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1312com12 32 . . . . . . . . 9 (𝑦𝑧 → (𝑧 ∈ On → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1413impd 412 . . . . . . . 8 (𝑦𝑧 → ((𝑧 ∈ On ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴)))
15 breq1 5152 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1615elrab 3684 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑧 ∈ On ∧ 𝑧𝐴))
17 breq1 5152 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1817elrab 3684 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
1914, 16, 183imtr4g 296 . . . . . . 7 (𝑦𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2019imp 408 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
2120gen2 1799 . . . . 5 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
22 dftr2 5268 . . . . 5 (Tr {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2321, 22mpbir 230 . . . 4 Tr {𝑥 ∈ On ∣ 𝑥𝐴}
24 ssrab2 4078 . . . 4 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On
25 ordon 7764 . . . 4 Ord On
26 trssord 6382 . . . 4 ((Tr {𝑥 ∈ On ∣ 𝑥𝐴} ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥𝐴})
2723, 24, 25, 26mp3an 1462 . . 3 Ord {𝑥 ∈ On ∣ 𝑥𝐴}
28 hartogs 9539 . . . 4 (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
29 sdomdom 8976 . . . . . . 7 (𝑥𝐴𝑥𝐴)
3029a1i 11 . . . . . 6 (𝑥 ∈ On → (𝑥𝐴𝑥𝐴))
3130ss2rabi 4075 . . . . 5 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥𝐴}
32 ssexg 5324 . . . . 5 (({𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥𝐴} ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On) → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
3331, 32mpan 689 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
34 elong 6373 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ V → ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥𝐴}))
3528, 33, 343syl 18 . . 3 (𝐴 ∈ V → ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥𝐴}))
3627, 35mpbiri 258 . 2 (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
37 0elon 6419 . . . 4 ∅ ∈ On
38 eleq1 2822 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} = ∅ → ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On ↔ ∅ ∈ On))
3937, 38mpbiri 258 . . 3 ({𝑥 ∈ On ∣ 𝑥𝐴} = ∅ → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
40 df-ne 2942 . . . . 5 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ¬ {𝑥 ∈ On ∣ 𝑥𝐴} = ∅)
41 rabn0 4386 . . . . 5 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
4240, 41bitr3i 277 . . . 4 (¬ {𝑥 ∈ On ∣ 𝑥𝐴} = ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
43 relsdom 8946 . . . . . 6 Rel ≺
4443brrelex2i 5734 . . . . 5 (𝑥𝐴𝐴 ∈ V)
4544rexlimivw 3152 . . . 4 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
4642, 45sylbi 216 . . 3 (¬ {𝑥 ∈ On ∣ 𝑥𝐴} = ∅ → 𝐴 ∈ V)
4739, 46nsyl4 158 . 2 𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
4836, 47pm2.61i 182 1 {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107  wne 2941  wrex 3071  {crab 3433  Vcvv 3475  wss 3949  c0 4323   class class class wbr 5149  Tr wtr 5266  Ord word 6364  Oncon0 6365  cdom 8937  csdm 8938
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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-oi 9505
This theorem is referenced by: (None)
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