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Theorem card2on 9044
 Description: The alternate definition of the cardinal of a set given in cardval2 9446 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 6195 . . . . . . . . . . . . 13 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
2 vex 3414 . . . . . . . . . . . . . 14 𝑧 ∈ V
3 onelss 6212 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
43imp 411 . . . . . . . . . . . . . 14 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
5 ssdomg 8574 . . . . . . . . . . . . . 14 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
62, 4, 5mpsyl 68 . . . . . . . . . . . . 13 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
71, 6jca 516 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → (𝑦 ∈ On ∧ 𝑦𝑧))
8 domsdomtr 8674 . . . . . . . . . . . . . 14 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
98anim2i 620 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ (𝑦𝑧𝑧𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
109anassrs 472 . . . . . . . . . . . 12 (((𝑦 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
117, 10sylan 584 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
1211exp31 424 . . . . . . . . . 10 (𝑧 ∈ On → (𝑦𝑧 → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1312com12 32 . . . . . . . . 9 (𝑦𝑧 → (𝑧 ∈ On → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1413impd 415 . . . . . . . 8 (𝑦𝑧 → ((𝑧 ∈ On ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴)))
15 breq1 5036 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1615elrab 3603 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑧 ∈ On ∧ 𝑧𝐴))
17 breq1 5036 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1817elrab 3603 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
1914, 16, 183imtr4g 300 . . . . . . 7 (𝑦𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2019imp 411 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
2120gen2 1799 . . . . 5 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
22 dftr2 5141 . . . . 5 (Tr {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2321, 22mpbir 234 . . . 4 Tr {𝑥 ∈ On ∣ 𝑥𝐴}
24 ssrab2 3985 . . . 4 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On
25 ordon 7498 . . . 4 Ord On
26 trssord 6187 . . . 4 ((Tr {𝑥 ∈ On ∣ 𝑥𝐴} ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥𝐴})
2723, 24, 25, 26mp3an 1459 . . 3 Ord {𝑥 ∈ On ∣ 𝑥𝐴}
28 hartogs 9034 . . . 4 (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
29 sdomdom 8556 . . . . . . 7 (𝑥𝐴𝑥𝐴)
3029a1i 11 . . . . . 6 (𝑥 ∈ On → (𝑥𝐴𝑥𝐴))
3130ss2rabi 3982 . . . . 5 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥𝐴}
32 ssexg 5194 . . . . 5 (({𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥𝐴} ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On) → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
3331, 32mpan 690 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
34 elong 6178 . . . 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 261 . 2 (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
37 0elon 6223 . . . 4 ∅ ∈ On
38 eleq1 2840 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} = ∅ → ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On ↔ ∅ ∈ On))
3937, 38mpbiri 261 . . 3 ({𝑥 ∈ On ∣ 𝑥𝐴} = ∅ → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
40 df-ne 2953 . . . . 5 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ¬ {𝑥 ∈ On ∣ 𝑥𝐴} = ∅)
41 rabn0 4282 . . . . 5 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
4240, 41bitr3i 280 . . . 4 (¬ {𝑥 ∈ On ∣ 𝑥𝐴} = ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
43 relsdom 8535 . . . . . 6 Rel ≺
4443brrelex2i 5579 . . . . 5 (𝑥𝐴𝐴 ∈ V)
4544rexlimivw 3207 . . . 4 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
4642, 45sylbi 220 . . 3 (¬ {𝑥 ∈ On ∣ 𝑥𝐴} = ∅ → 𝐴 ∈ V)
4739, 46nsyl4 161 . 2 𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
4836, 47pm2.61i 185 1 {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∃wrex 3072  {crab 3075  Vcvv 3410   ⊆ wss 3859  ∅c0 4226   class class class wbr 5033  Tr wtr 5139  Ord word 6169  Oncon0 6170   ≼ cdom 8526   ≺ csdm 8527 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-se 5485  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7109  df-wrecs 7958  df-recs 8019  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-oi 9000 This theorem is referenced by: (None)
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