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Mirrors > Home > MPE Home > Th. List > alephprc | Structured version Visualization version GIF version |
Description: The class of all transfinite cardinal numbers (the range of the aleph function) is a proper class. Proposition 10.26 of [TakeutiZaring] p. 90. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
alephprc | ⊢ ¬ ran ℵ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardprc 8806 | . . . 4 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V | |
2 | 1 | neli 2899 | . . 3 ⊢ ¬ {𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V |
3 | cardnum 8917 | . . . 4 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} = (ω ∪ ran ℵ) | |
4 | 3 | eleq1i 2692 | . . 3 ⊢ ({𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V ↔ (ω ∪ ran ℵ) ∈ V) |
5 | 2, 4 | mtbi 312 | . 2 ⊢ ¬ (ω ∪ ran ℵ) ∈ V |
6 | omex 8540 | . . 3 ⊢ ω ∈ V | |
7 | unexg 6959 | . . 3 ⊢ ((ω ∈ V ∧ ran ℵ ∈ V) → (ω ∪ ran ℵ) ∈ V) | |
8 | 6, 7 | mpan 706 | . 2 ⊢ (ran ℵ ∈ V → (ω ∪ ran ℵ) ∈ V) |
9 | 5, 8 | mto 188 | 1 ⊢ ¬ ran ℵ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 {cab 2608 Vcvv 3200 ∪ cun 3572 ran crn 5115 ‘cfv 5888 ωcom 7065 cardccrd 8761 ℵcale 8762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-har 8463 df-card 8765 df-aleph 8766 |
This theorem is referenced by: unialeph 8924 |
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