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Mirrors > Home > MPE Home > Th. List > gch-kn | Structured version Visualization version GIF version |
Description: The equivalence of two versions of the Generalized Continuum Hypothesis. The right-hand side is the standard version in the literature. The left-hand side is a version devised by Kannan Nambiar, which he calls the Axiom of Combinatorial Sets. For the notation and motivation behind this axiom, see his paper, "Derivation of Continuum Hypothesis from Axiom of Combinatorial Sets", available at http://www.e-atheneum.net/science/derivation_ch.pdf. The equivalence of the two sides provides a negative answer to Open Problem 2 in http://www.e-atheneum.net/science/open_problem_print.pdf. The key idea in the proof below is to equate both sides of alephexp2 10598 to the successor aleph using enen2 9136. (Contributed by NM, 1-Oct-2004.) |
Ref | Expression |
---|---|
gch-kn | ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (ℵ‘suc 𝐴) ≈ (2o ↑m (ℵ‘𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephexp2 10598 | . . 3 ⊢ (𝐴 ∈ On → (2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}) | |
2 | enen2 9136 | . . 3 ⊢ ((2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} → ((ℵ‘suc 𝐴) ≈ (2o ↑m (ℵ‘𝐴)) ↔ (ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ (2o ↑m (ℵ‘𝐴)) ↔ (ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})) |
4 | 3 | bicomd 222 | 1 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (ℵ‘suc 𝐴) ≈ (2o ↑m (ℵ‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2099 {cab 2704 ⊆ wss 3944 class class class wbr 5142 Oncon0 6363 suc csuc 6365 ‘cfv 6542 (class class class)co 7414 2oc2o 8474 ↑m cmap 8838 ≈ cen 8954 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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-ac2 10480 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-oi 9527 df-har 9574 df-card 9956 df-aleph 9957 df-acn 9959 df-ac 10133 |
This theorem is referenced by: (None) |
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