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Mirrors > Home > MPE Home > Th. List > alephfp2 | Structured version Visualization version GIF version |
Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 9608 for an actual example of a fixed point. Compare the inequality alephle 9588 that holds in general. Note that if 𝑥 is a fixed point, then ℵ‘ℵ‘ℵ‘... ℵ‘𝑥 = 𝑥. (Contributed by NM, 6-Nov-2004.) (Revised by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
alephfp2 | ⊢ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephsson 9600 | . . 3 ⊢ ran ℵ ⊆ On | |
2 | eqid 2738 | . . . 4 ⊢ (rec(ℵ, ω) ↾ ω) = (rec(ℵ, ω) ↾ ω) | |
3 | 2 | alephfplem4 9607 | . . 3 ⊢ ∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ ran ℵ |
4 | 1, 3 | sselii 3874 | . 2 ⊢ ∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On |
5 | 2 | alephfp 9608 | . 2 ⊢ (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)) = ∪ ((rec(ℵ, ω) ↾ ω) “ ω) |
6 | fveq2 6674 | . . . 4 ⊢ (𝑥 = ∪ ((rec(ℵ, ω) ↾ ω) “ ω) → (ℵ‘𝑥) = (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω))) | |
7 | id 22 | . . . 4 ⊢ (𝑥 = ∪ ((rec(ℵ, ω) ↾ ω) “ ω) → 𝑥 = ∪ ((rec(ℵ, ω) ↾ ω) “ ω)) | |
8 | 6, 7 | eqeq12d 2754 | . . 3 ⊢ (𝑥 = ∪ ((rec(ℵ, ω) ↾ ω) “ ω) → ((ℵ‘𝑥) = 𝑥 ↔ (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)) = ∪ ((rec(ℵ, ω) ↾ ω) “ ω))) |
9 | 8 | rspcev 3526 | . 2 ⊢ ((∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On ∧ (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)) = ∪ ((rec(ℵ, ω) ↾ ω) “ ω)) → ∃𝑥 ∈ On (ℵ‘𝑥) = 𝑥) |
10 | 4, 5, 9 | mp2an 692 | 1 ⊢ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 ∃wrex 3054 ∪ cuni 4796 ran crn 5526 ↾ cres 5527 “ cima 5528 Oncon0 6172 ‘cfv 6339 ωcom 7599 reccrdg 8074 ℵcale 9438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-oi 9047 df-har 9094 df-card 9441 df-aleph 9442 |
This theorem is referenced by: (None) |
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