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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 9865 | . . . 4 ⊢ ¬ ran ℵ ∈ V | |
2 | uniexb 7604 | . . . 4 ⊢ (ran ℵ ∈ V ↔ ∪ ran ℵ ∈ V) | |
3 | 1, 2 | mtbi 322 | . . 3 ⊢ ¬ ∪ ran ℵ ∈ V |
4 | elex 3447 | . . 3 ⊢ (∪ ran ℵ ∈ On → ∪ ran ℵ ∈ V) | |
5 | 3, 4 | mto 196 | . 2 ⊢ ¬ ∪ ran ℵ ∈ On |
6 | alephsson 9866 | . . . 4 ⊢ ran ℵ ⊆ On | |
7 | ssorduni 7619 | . . . 4 ⊢ (ran ℵ ⊆ On → Ord ∪ ran ℵ) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ Ord ∪ ran ℵ |
9 | ordeleqon 7622 | . . 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 1773 | 1 ⊢ ∪ ran ℵ = On |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 844 = wceq 1539 ∈ wcel 2106 Vcvv 3429 ⊆ wss 3886 ∪ cuni 4839 ran crn 5585 Ord word 6258 Oncon0 6259 ℵcale 9704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-om 7703 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-oi 9256 df-har 9303 df-card 9707 df-aleph 9708 |
This theorem is referenced by: (None) |
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