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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 4476 | . . . . 5 ⊢ ((({∅} × 𝐴) ∪ ({1o} × 1o)) ∖ ({1o} × 1o)) = (({∅} × 𝐴) ∖ ({1o} × 1o)) | |
2 | df-dju 9918 | . . . . . 6 ⊢ (𝐴 ⊔ 1o) = (({∅} × 𝐴) ∪ ({1o} × 1o)) | |
3 | df1o2 8487 | . . . . . . . 8 ⊢ 1o = {∅} | |
4 | 3 | xpeq2i 5699 | . . . . . . 7 ⊢ ({1o} × 1o) = ({1o} × {∅}) |
5 | 1oex 8490 | . . . . . . . 8 ⊢ 1o ∈ V | |
6 | 0ex 5301 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 5, 6 | xpsn 7144 | . . . . . . 7 ⊢ ({1o} × {∅}) = {〈1o, ∅〉} |
8 | 4, 7 | eqtr2i 2756 | . . . . . 6 ⊢ {〈1o, ∅〉} = ({1o} × 1o) |
9 | 2, 8 | difeq12i 4116 | . . . . 5 ⊢ ((𝐴 ⊔ 1o) ∖ {〈1o, ∅〉}) = ((({∅} × 𝐴) ∪ ({1o} × 1o)) ∖ ({1o} × 1o)) |
10 | xp01disjl 8506 | . . . . . 6 ⊢ (({∅} × 𝐴) ∩ ({1o} × 1o)) = ∅ | |
11 | disj3 4449 | . . . . . 6 ⊢ ((({∅} × 𝐴) ∩ ({1o} × 1o)) = ∅ ↔ ({∅} × 𝐴) = (({∅} × 𝐴) ∖ ({1o} × 1o))) | |
12 | 10, 11 | mpbi 229 | . . . . 5 ⊢ ({∅} × 𝐴) = (({∅} × 𝐴) ∖ ({1o} × 1o)) |
13 | 1, 9, 12 | 3eqtr4i 2765 | . . . 4 ⊢ ((𝐴 ⊔ 1o) ∖ {〈1o, ∅〉}) = ({∅} × 𝐴) |
14 | reldom 8963 | . . . . . . . 8 ⊢ Rel ≼ | |
15 | 14 | brrelex2i 5729 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
16 | 1on 8492 | . . . . . . 7 ⊢ 1o ∈ On | |
17 | djudoml 10201 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 1o ∈ On) → 𝐴 ≼ (𝐴 ⊔ 1o)) | |
18 | 15, 16, 17 | sylancl 585 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 ⊔ 1o)) |
19 | domtr 9021 | . . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ⊔ 1o)) → ω ≼ (𝐴 ⊔ 1o)) | |
20 | 18, 19 | mpdan 686 | . . . . 5 ⊢ (ω ≼ 𝐴 → ω ≼ (𝐴 ⊔ 1o)) |
21 | infdifsn 9674 | . . . . 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 5178 | . . 3 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ (𝐴 ⊔ 1o)) |
24 | 23 | ensymd 9019 | . 2 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 1o) ≈ ({∅} × 𝐴)) |
25 | xpsnen2g 9083 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
26 | 6, 15, 25 | sylancr 586 | . 2 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
27 | entr 9020 | . 2 ⊢ (((𝐴 ⊔ 1o) ≈ ({∅} × 𝐴) ∧ ({∅} × 𝐴) ≈ 𝐴) → (𝐴 ⊔ 1o) ≈ 𝐴) | |
28 | 24, 26, 27 | syl2anc 583 | 1 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 1o) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 ∅c0 4318 {csn 4624 〈cop 4630 class class class wbr 5142 × cxp 5670 Oncon0 6363 ωcom 7864 1oc1o 8473 ≈ cen 8954 ≼ cdom 8955 ⊔ cdju 9915 |
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-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
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-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-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-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-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-om 7865 df-1st 7987 df-2nd 7988 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-dju 9918 |
This theorem is referenced by: pwdjuidm 10208 isfin4p1 10332 canthp1lem2 10670 |
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