Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > infdju1 | Structured version Visualization version GIF version |
Description: An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
infdju1 | ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 1o) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difun2 4420 | . . . . 5 ⊢ ((({∅} × 𝐴) ∪ ({1o} × 1o)) ∖ ({1o} × 1o)) = (({∅} × 𝐴) ∖ ({1o} × 1o)) | |
2 | df-dju 9703 | . . . . . 6 ⊢ (𝐴 ⊔ 1o) = (({∅} × 𝐴) ∪ ({1o} × 1o)) | |
3 | df1o2 8335 | . . . . . . . 8 ⊢ 1o = {∅} | |
4 | 3 | xpeq2i 5627 | . . . . . . 7 ⊢ ({1o} × 1o) = ({1o} × {∅}) |
5 | 1oex 8338 | . . . . . . . 8 ⊢ 1o ∈ V | |
6 | 0ex 5240 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 5, 6 | xpsn 7045 | . . . . . . 7 ⊢ ({1o} × {∅}) = {⟨1o, ∅⟩} |
8 | 4, 7 | eqtr2i 2765 | . . . . . 6 ⊢ {⟨1o, ∅⟩} = ({1o} × 1o) |
9 | 2, 8 | difeq12i 4061 | . . . . 5 ⊢ ((𝐴 ⊔ 1o) ∖ {⟨1o, ∅⟩}) = ((({∅} × 𝐴) ∪ ({1o} × 1o)) ∖ ({1o} × 1o)) |
10 | xp01disjl 8353 | . . . . . 6 ⊢ (({∅} × 𝐴) ∩ ({1o} × 1o)) = ∅ | |
11 | disj3 4393 | . . . . . 6 ⊢ ((({∅} × 𝐴) ∩ ({1o} × 1o)) = ∅ ↔ ({∅} × 𝐴) = (({∅} × 𝐴) ∖ ({1o} × 1o))) | |
12 | 10, 11 | mpbi 229 | . . . . 5 ⊢ ({∅} × 𝐴) = (({∅} × 𝐴) ∖ ({1o} × 1o)) |
13 | 1, 9, 12 | 3eqtr4i 2774 | . . . 4 ⊢ ((𝐴 ⊔ 1o) ∖ {⟨1o, ∅⟩}) = ({∅} × 𝐴) |
14 | reldom 8770 | . . . . . . . 8 ⊢ Rel ≼ | |
15 | 14 | brrelex2i 5655 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
16 | 1on 8340 | . . . . . . 7 ⊢ 1o ∈ On | |
17 | djudoml 9986 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 1o ∈ On) → 𝐴 ≼ (𝐴 ⊔ 1o)) | |
18 | 15, 16, 17 | sylancl 587 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 ⊔ 1o)) |
19 | domtr 8828 | . . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ⊔ 1o)) → ω ≼ (𝐴 ⊔ 1o)) | |
20 | 18, 19 | mpdan 685 | . . . . 5 ⊢ (ω ≼ 𝐴 → ω ≼ (𝐴 ⊔ 1o)) |
21 | infdifsn 9459 | . . . . 5 ⊢ (ω ≼ (𝐴 ⊔ 1o) → ((𝐴 ⊔ 1o) ∖ {⟨1o, ∅⟩}) ≈ (𝐴 ⊔ 1o)) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (ω ≼ 𝐴 → ((𝐴 ⊔ 1o) ∖ {⟨1o, ∅⟩}) ≈ (𝐴 ⊔ 1o)) |
23 | 13, 22 | eqbrtrrid 5117 | . . 3 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ (𝐴 ⊔ 1o)) |
24 | 23 | ensymd 8826 | . 2 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 1o) ≈ ({∅} × 𝐴)) |
25 | xpsnen2g 8890 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
26 | 6, 15, 25 | sylancr 588 | . 2 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
27 | entr 8827 | . 2 ⊢ (((𝐴 ⊔ 1o) ≈ ({∅} × 𝐴) ∧ ({∅} × 𝐴) ≈ 𝐴) → (𝐴 ⊔ 1o) ≈ 𝐴) | |
28 | 24, 26, 27 | syl2anc 585 | 1 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 1o) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 Vcvv 3437 ∖ cdif 3889 ∪ cun 3890 ∩ cin 3891 ∅c0 4262 {csn 4565 ⟨cop 4571 class class class wbr 5081 × cxp 5598 Oncon0 6281 ωcom 7744 1oc1o 8321 ≈ cen 8761 ≼ cdom 8762 ⊔ cdju 9700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-om 7745 df-1st 7863 df-2nd 7864 df-1o 8328 df-er 8529 df-en 8765 df-dom 8766 df-dju 9703 |
This theorem is referenced by: pwdjuidm 9993 isfin4p1 10117 canthp1lem2 10455 |
Copyright terms: Public domain | W3C validator |