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Theorem card2on 9498
Description: The alternate definition of the cardinal of a set given in cardval2 9935 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 6346 . . . . . . . . . . . . 13 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
2 vex 3451 . . . . . . . . . . . . . 14 𝑧 ∈ V
3 onelss 6363 . . . . . . . . . . . . . . 15 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
43imp 408 . . . . . . . . . . . . . 14 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
5 ssdomg 8946 . . . . . . . . . . . . . 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 9062 . . . . . . . . . . . . . 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 5112 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1615elrab 3649 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑧 ∈ On ∧ 𝑧𝐴))
17 breq1 5112 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1817elrab 3649 . . . . . . . 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 5228 . . . . 5 (Tr {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2321, 22mpbir 230 . . . 4 Tr {𝑥 ∈ On ∣ 𝑥𝐴}
24 ssrab2 4041 . . . 4 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On
25 ordon 7715 . . . 4 Ord On
26 trssord 6338 . . . 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 9488 . . . 4 (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
29 sdomdom 8926 . . . . . . 7 (𝑥𝐴𝑥𝐴)
3029a1i 11 . . . . . 6 (𝑥 ∈ On → (𝑥𝐴𝑥𝐴))
3130ss2rabi 4038 . . . . 5 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥𝐴}
32 ssexg 5284 . . . . 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 6329 . . . 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 6375 . . . 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 2941 . . . . 5 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ¬ {𝑥 ∈ On ∣ 𝑥𝐴} = ∅)
41 rabn0 4349 . . . . 5 ({𝑥 ∈ On ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
4240, 41bitr3i 277 . . . 4 (¬ {𝑥 ∈ On ∣ 𝑥𝐴} = ∅ ↔ ∃𝑥 ∈ On 𝑥𝐴)
43 relsdom 8896 . . . . . 6 Rel ≺
4443brrelex2i 5693 . . . . 5 (𝑥𝐴𝐴 ∈ V)
4544rexlimivw 3145 . . . 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 2940  wrex 3070  {crab 3406  Vcvv 3447  wss 3914  c0 4286   class class class wbr 5109  Tr wtr 5226  Ord word 6320  Oncon0 6321  cdom 8887  csdm 8888
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-oi 9454
This theorem is referenced by: (None)
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