Theorem unialeph 9788

Description: The union of the class of transfinite cardinals (the range of the aleph function) is the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)

Assertion

Ref	Expression
unialeph	⊢ ∪ ran ℵ = On

Proof of Theorem unialeph

Step	Hyp	Ref	Expression
1		alephprc 9786	. . . 4 ⊢ ¬ ran ℵ ∈ V
2		uniexb 7592	. . . 4 ⊢ (ran ℵ ∈ V ↔ ∪ ran ℵ ∈ V)
3	1, 2	mtbi 321	. . 3 ⊢ ¬ ∪ ran ℵ ∈ V
4		elex 3440	. . 3 ⊢ (∪ ran ℵ ∈ On → ∪ ran ℵ ∈ V)
5	3, 4	mto 196	. 2 ⊢ ¬ ∪ ran ℵ ∈ On
6		alephsson 9787	. . . 4 ⊢ ran ℵ ⊆ On
7		ssorduni 7606	. . . 4 ⊢ (ran ℵ ⊆ On → Ord ∪ ran ℵ)
8	6, 7	ax-mp 5	. . 3 ⊢ Ord ∪ ran ℵ
9		ordeleqon 7609	. . 3 ⊢ (Ord ∪ ran ℵ ↔ (∪ ran ℵ ∈ On ∨ ∪ ran ℵ = On))
10	8, 9	mpbi 229	. 2 ⊢ (∪ ran ℵ ∈ On ∨ ∪ ran ℵ = On)
11	5, 10	mtpor 1774	1 ⊢ ∪ ran ℵ = On

Syntax hints:   ∨ wo 843   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  ∪ cuni 4836  ran crn 5581  Ord word 6250  Oncon0 6251  ℵcale 9625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-har 9246  df-card 9628  df-aleph 9629

This theorem is referenced by: (None)