Theorem card2on 9523

Description: The alternate definition of the cardinal of a set given in cardval2 9993 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 3461	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
3		onelss 6407	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
4	3	imp 412	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
5		ssdomg 9003	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (𝑦 ⊆ 𝑧 → 𝑦 ≼ 𝑧))
6	2, 4, 5	mpsyl 69	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ≼ 𝑧)
7	1, 6	jca 521	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝑧))
8		domsdomtr 9107	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≼ 𝑧 ∧ 𝑧 ≺ 𝐴) → 𝑦 ≺ 𝐴)
9	8	anim2i 629	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ (𝑦 ≼ 𝑧 ∧ 𝑧 ≺ 𝐴)) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
10	9	anassrs 473	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≼ 𝑧) ∧ 𝑧 ≺ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
11	7, 10	sylan 592	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) ∧ 𝑧 ≺ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
12	11	exp31 425	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → (𝑧 ≺ 𝐴 → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))))
13	12	com12 33	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑧 → (𝑧 ∈ On → (𝑧 ≺ 𝐴 → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))))
14	13	impd 416	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑧 → ((𝑧 ∈ On ∧ 𝑧 ≺ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴)))
15		breq1 5114	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
16	15	elrab 3652	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (𝑧 ∈ On ∧ 𝑧 ≺ 𝐴))
17		breq1 5114	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝐴 ↔ 𝑦 ≺ 𝐴))
18	17	elrab 3652	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
19	14, 16, 18	3imtr4g 299	. . . . . . 7 ⊢ (𝑦 ∈ 𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
20	19	imp 412	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
21	20	gen2 1829	. . . . 5 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
22		dftr2 5222	. . . . 5 ⊢ (Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
23	21, 22	mpbir 234	. . . 4 ⊢ Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}
24		ssrab2 4035	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ On
25		ordon 7782	. . . 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 1490	. . 3 ⊢ Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}
28		hartogs 9513	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)
29		sdomdom 8983	. . . . . . 7 ⊢ (𝑥 ≺ 𝐴 → 𝑥 ≼ 𝐴)
30	29	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ On → (𝑥 ≺ 𝐴 → 𝑥 ≼ 𝐴))
31	30	ss2rabi 4031	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
32		ssexg 5292	. . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On) → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ V)
33	31, 32	mpan 703	. . . 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 19	. . 3 ⊢ (𝐴 ∈ V → ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
36	27, 35	mpbiri 261	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On)
37		0elon 6420	. . . 4 ⊢ ∅ ∈ On
38		eleq1 2853	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ → ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On ↔ ∅ ∈ On))
39	37, 38	mpbiri 261	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On)
40		df-ne 2961	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ≠ ∅ ↔ ¬ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅)
41		rabn0 4346	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥 ≺ 𝐴)
42	40, 41	bitr3i 280	. . . 4 ⊢ (¬ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ ↔ ∃𝑥 ∈ On 𝑥 ≺ 𝐴)
43		relsdom 8956	. . . . . 6 ⊢ Rel ≺
44	43	brrelex2i 5720	. . . . 5 ⊢ (𝑥 ≺ 𝐴 → 𝐴 ∈ V)
45	44	rexlimivw 3164	. . . 4 ⊢ (∃𝑥 ∈ On 𝑥 ≺ 𝐴 → 𝐴 ∈ V)
46	42, 45	sylbi 220	. . 3 ⊢ (¬ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ → 𝐴 ∈ V)
47	39, 46	nsyl4 159	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On)
48	36, 47	pm2.61i 184	1 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091  {crab 3418  Vcvv 3457   ⊆ wss 3906  ∅c0 4286   class class class wbr 5111  Tr wtr 5220  Ord word 6363  Oncon0 6364   ≼ cdom 8947   ≺ csdm 8948

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-oi 9479

This theorem is used by: (None)