Theorem card2on 9440

Description: The alternate definition of the cardinal of a set given in cardval2 9884 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 6331	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
2		vex 3440	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
3		onelss 6348	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
4	3	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
5		ssdomg 8922	. . . . . . . . . . . . . 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 9025	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≼ 𝑧 ∧ 𝑧 ≺ 𝐴) → 𝑦 ≺ 𝐴)
9	8	anim2i 617	. . . . . . . . . . . . 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 580	. . . . . . . . . . 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 5092	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
16	15	elrab 3642	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (𝑧 ∈ On ∧ 𝑧 ≺ 𝐴))
17		breq1 5092	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝐴 ↔ 𝑦 ≺ 𝐴))
18	17	elrab 3642	. . . . . . . 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 5198	. . . . 5 ⊢ (Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
23	21, 22	mpbir 231	. . . 4 ⊢ Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}
24		ssrab2 4027	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ On
25		ordon 7710	. . . 4 ⊢ Ord On
26		trssord 6323	. . . 4 ⊢ ((Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
27	23, 24, 25, 26	mp3an 1463	. . 3 ⊢ Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}
28		hartogs 9430	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)
29		sdomdom 8902	. . . . . . 7 ⊢ (𝑥 ≺ 𝐴 → 𝑥 ≼ 𝐴)
30	29	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ On → (𝑥 ≺ 𝐴 → 𝑥 ≼ 𝐴))
31	30	ss2rabi 4023	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
32		ssexg 5259	. . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On) → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ V)
33	31, 32	mpan 690	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ V)
34		elong 6314	. . . 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 6361	. . . 4 ⊢ ∅ ∈ On
38		eleq1 2819	. . . 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 2929	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ≠ ∅ ↔ ¬ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅)
41		rabn0 4336	. . . . 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 8876	. . . . . 6 ⊢ Rel ≺
44	43	brrelex2i 5671	. . . . 5 ⊢ (𝑥 ≺ 𝐴 → 𝐴 ∈ V)
45	44	rexlimivw 3129	. . . 4 ⊢ (∃𝑥 ∈ On 𝑥 ≺ 𝐴 → 𝐴 ∈ V)
46	42, 45	sylbi 217	. . 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 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  {crab 3395  Vcvv 3436   ⊆ wss 3897  ∅c0 4280   class class class wbr 5089  Tr wtr 5196  Ord word 6305  Oncon0 6306   ≼ cdom 8867   ≺ csdm 8868

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-oi 9396

This theorem is referenced by: (None)