Theorem card2on 9555

Description: The alternate definition of the cardinal of a set given in cardval2 9992 always gives a set, and indeed an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)

Assertion

Ref	Expression
card2on	⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On

Distinct variable group:   𝑥,𝐴

Proof of Theorem card2on

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onelon 6389	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
2		vex 3477	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
3		onelss 6406	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
4	3	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
5		ssdomg 9002	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (𝑦 ⊆ 𝑧 → 𝑦 ≼ 𝑧))
6	2, 4, 5	mpsyl 68	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ≼ 𝑧)
7	1, 6	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝑧))
8		domsdomtr 9118	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≼ 𝑧 ∧ 𝑧 ≺ 𝐴) → 𝑦 ≺ 𝐴)
9	8	anim2i 616	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ (𝑦 ≼ 𝑧 ∧ 𝑧 ≺ 𝐴)) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
10	9	anassrs 467	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≼ 𝑧) ∧ 𝑧 ≺ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
11	7, 10	sylan 579	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) ∧ 𝑧 ≺ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
12	11	exp31 419	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → (𝑧 ≺ 𝐴 → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))))
13	12	com12 32	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑧 → (𝑧 ∈ On → (𝑧 ≺ 𝐴 → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))))
14	13	impd 410	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑧 → ((𝑧 ∈ On ∧ 𝑧 ≺ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴)))
15		breq1 5151	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
16	15	elrab 3683	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (𝑧 ∈ On ∧ 𝑧 ≺ 𝐴))
17		breq1 5151	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝐴 ↔ 𝑦 ≺ 𝐴))
18	17	elrab 3683	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
19	14, 16, 18	3imtr4g 296	. . . . . . 7 ⊢ (𝑦 ∈ 𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
20	19	imp 406	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
21	20	gen2 1797	. . . . 5 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
22		dftr2 5267	. . . . 5 ⊢ (Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
23	21, 22	mpbir 230	. . . 4 ⊢ Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}
24		ssrab2 4077	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ On
25		ordon 7768	. . . 4 ⊢ Ord On
26		trssord 6381	. . . 4 ⊢ ((Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
27	23, 24, 25, 26	mp3an 1460	. . 3 ⊢ Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}
28		hartogs 9545	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)
29		sdomdom 8982	. . . . . . 7 ⊢ (𝑥 ≺ 𝐴 → 𝑥 ≼ 𝐴)
30	29	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ On → (𝑥 ≺ 𝐴 → 𝑥 ≼ 𝐴))
31	30	ss2rabi 4074	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
32		ssexg 5323	. . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On) → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ V)
33	31, 32	mpan 687	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ V)
34		elong 6372	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ V → ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
35	28, 33, 34	3syl 18	. . 3 ⊢ (𝐴 ∈ V → ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
36	27, 35	mpbiri 258	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On)
37		0elon 6418	. . . 4 ⊢ ∅ ∈ On
38		eleq1 2820	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ → ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On ↔ ∅ ∈ On))
39	37, 38	mpbiri 258	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On)
40		df-ne 2940	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ≠ ∅ ↔ ¬ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅)
41		rabn0 4385	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥 ≺ 𝐴)
42	40, 41	bitr3i 277	. . . 4 ⊢ (¬ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ ↔ ∃𝑥 ∈ On 𝑥 ≺ 𝐴)
43		relsdom 8952	. . . . . 6 ⊢ Rel ≺
44	43	brrelex2i 5733	. . . . 5 ⊢ (𝑥 ≺ 𝐴 → 𝐴 ∈ V)
45	44	rexlimivw 3150	. . . 4 ⊢ (∃𝑥 ∈ On 𝑥 ≺ 𝐴 → 𝐴 ∈ V)
46	42, 45	sylbi 216	. . 3 ⊢ (¬ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ → 𝐴 ∈ V)
47	39, 46	nsyl4 158	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On)
48	36, 47	pm2.61i 182	1 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∃wrex 3069  {crab 3431  Vcvv 3473   ⊆ wss 3948  ∅c0 4322   class class class wbr 5148  Tr wtr 5265  Ord word 6363  Oncon0 6364   ≼ cdom 8943   ≺ csdm 8944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-oi 9511

This theorem is referenced by: (None)