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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 9016 | . . . 4 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V | |
2 | 1 | neli 3037 | . . 3 ⊢ ¬ {𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V |
3 | cardnum 9127 | . . . 4 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} = (ω ∪ ran ℵ) | |
4 | 3 | eleq1i 2830 | . . 3 ⊢ ({𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V ↔ (ω ∪ ran ℵ) ∈ V) |
5 | 2, 4 | mtbi 311 | . 2 ⊢ ¬ (ω ∪ ran ℵ) ∈ V |
6 | omex 8715 | . . 3 ⊢ ω ∈ V | |
7 | unexg 7125 | . . 3 ⊢ ((ω ∈ V ∧ ran ℵ ∈ V) → (ω ∪ ran ℵ) ∈ V) | |
8 | 6, 7 | mpan 708 | . 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 1632 ∈ wcel 2139 {cab 2746 Vcvv 3340 ∪ cun 3713 ran crn 5267 ‘cfv 6049 ωcom 7231 cardccrd 8971 ℵcale 8972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-om 7232 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-oi 8582 df-har 8630 df-card 8975 df-aleph 8976 |
This theorem is referenced by: unialeph 9134 |
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