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Theorem card2on 9592
Description: The alternate definition of the cardinal of a set given in cardval2 10029 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 6411 . . . . . . . . . . . . 13 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
2 vex 3482 . . . . . . . . . . . . . 14 𝑧 ∈ V
3 onelss 6428 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
43imp 406 . . . . . . . . . . . . . 14 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
5 ssdomg 9039 . . . . . . . . . . . . . 14 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
62, 4, 5mpsyl 68 . . . . . . . . . . . . 13 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
71, 6jca 511 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → (𝑦 ∈ On ∧ 𝑦𝑧))
8 domsdomtr 9151 . . . . . . . . . . . . . 14 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
98anim2i 617 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ (𝑦𝑧𝑧𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
109anassrs 467 . . . . . . . . . . . 12 (((𝑦 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
117, 10sylan 580 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
1211exp31 419 . . . . . . . . . 10 (𝑧 ∈ On → (𝑦𝑧 → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1312com12 32 . . . . . . . . 9 (𝑦𝑧 → (𝑧 ∈ On → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1413impd 410 . . . . . . . 8 (𝑦𝑧 → ((𝑧 ∈ On ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴)))
15 breq1 5151 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1615elrab 3695 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑧 ∈ On ∧ 𝑧𝐴))
17 breq1 5151 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1817elrab 3695 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
1914, 16, 183imtr4g 296 . . . . . . 7 (𝑦𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2019imp 406 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
2120gen2 1793 . . . . 5 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
22 dftr2 5267 . . . . 5 (Tr {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2321, 22mpbir 231 . . . 4 Tr {𝑥 ∈ On ∣ 𝑥𝐴}
24 ssrab2 4090 . . . 4 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On
25 ordon 7796 . . . 4 Ord On
26 trssord 6403 . . . 4 ((Tr {𝑥 ∈ On ∣ 𝑥𝐴} ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥𝐴})
2723, 24, 25, 26mp3an 1460 . . 3 Ord {𝑥 ∈ On ∣ 𝑥𝐴}
28 hartogs 9582 . . . 4 (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
29 sdomdom 9019 . . . . . . 7 (𝑥𝐴𝑥𝐴)
3029a1i 11 . . . . . 6 (𝑥 ∈ On → (𝑥𝐴𝑥𝐴))
3130ss2rabi 4087 . . . . 5 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥𝐴}
32 ssexg 5329 . . . . 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 6394 . . . 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 6440 . . . 4 ∅ ∈ On
38 eleq1 2827 . . . 4 ({𝑥 ∈ On ∣ 𝑥𝐴} = ∅ → ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On ↔ ∅ ∈ On))
3937, 38mpbiri 258 . . 3 ({𝑥 ∈ On ∣ 𝑥𝐴} = ∅ → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
40 df-ne 2939 . . . . 5 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ¬ {𝑥 ∈ On ∣ 𝑥𝐴} = ∅)
41 rabn0 4395 . . . . 5 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
4240, 41bitr3i 277 . . . 4 (¬ {𝑥 ∈ On ∣ 𝑥𝐴} = ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
43 relsdom 8991 . . . . . 6 Rel ≺
4443brrelex2i 5746 . . . . 5 (𝑥𝐴𝐴 ∈ V)
4544rexlimivw 3149 . . . 4 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
4642, 45sylbi 217 . . 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 206  wa 395  wal 1535   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339   class class class wbr 5148  Tr wtr 5265  Ord word 6385  Oncon0 6386  cdom 8982  csdm 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-oi 9548
This theorem is referenced by: (None)
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