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Mirrors > Home > MPE Home > Th. List > unialeph | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
unialeph | ⊢ ∪ ran ℵ = On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephprc 10116 | . . . 4 ⊢ ¬ ran ℵ ∈ V | |
2 | uniexb 7760 | . . . 4 ⊢ (ran ℵ ∈ V ↔ ∪ ran ℵ ∈ V) | |
3 | 1, 2 | mtbi 322 | . . 3 ⊢ ¬ ∪ ran ℵ ∈ V |
4 | elex 3489 | . . 3 ⊢ (∪ ran ℵ ∈ On → ∪ ran ℵ ∈ V) | |
5 | 3, 4 | mto 196 | . 2 ⊢ ¬ ∪ ran ℵ ∈ On |
6 | alephsson 10117 | . . . 4 ⊢ ran ℵ ⊆ On | |
7 | ssorduni 7775 | . . . 4 ⊢ (ran ℵ ⊆ On → Ord ∪ ran ℵ) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ Ord ∪ ran ℵ |
9 | ordeleqon 7778 | . . 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 1765 | 1 ⊢ ∪ ran ℵ = On |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 846 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ⊆ wss 3945 ∪ cuni 4903 ran crn 5673 Ord word 6362 Oncon0 6363 ℵcale 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-oi 9527 df-har 9574 df-card 9956 df-aleph 9957 |
This theorem is referenced by: (None) |
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