Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > infcntss | Structured version Visualization version GIF version |
Description: Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.) |
Ref | Expression |
---|---|
infcntss.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
infcntss | ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infcntss.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | domen 7854 | . 2 ⊢ (ω ≼ 𝐴 ↔ ∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) |
3 | ensym 7891 | . . . . 5 ⊢ (ω ≈ 𝑥 → 𝑥 ≈ ω) | |
4 | 3 | anim2i 591 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ ω ≈ 𝑥) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω)) |
5 | 4 | ancoms 468 | . . 3 ⊢ ((ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω)) |
6 | 5 | eximi 1752 | . 2 ⊢ (∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω)) |
7 | 2, 6 | sylbi 206 | 1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-er 7629 df-en 7842 df-dom 7843 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |