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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 4484 | . . . . 5 ⊢ ((({∅} × 𝐴) ∪ ({1o} × 1o)) ∖ ({1o} × 1o)) = (({∅} × 𝐴) ∖ ({1o} × 1o)) | |
2 | df-dju 9932 | . . . . . 6 ⊢ (𝐴 ⊔ 1o) = (({∅} × 𝐴) ∪ ({1o} × 1o)) | |
3 | df1o2 8500 | . . . . . . . 8 ⊢ 1o = {∅} | |
4 | 3 | xpeq2i 5709 | . . . . . . 7 ⊢ ({1o} × 1o) = ({1o} × {∅}) |
5 | 1oex 8503 | . . . . . . . 8 ⊢ 1o ∈ V | |
6 | 0ex 5311 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 5, 6 | xpsn 7156 | . . . . . . 7 ⊢ ({1o} × {∅}) = {⟨1o, ∅⟩} |
8 | 4, 7 | eqtr2i 2757 | . . . . . 6 ⊢ {⟨1o, ∅⟩} = ({1o} × 1o) |
9 | 2, 8 | difeq12i 4120 | . . . . 5 ⊢ ((𝐴 ⊔ 1o) ∖ {⟨1o, ∅⟩}) = ((({∅} × 𝐴) ∪ ({1o} × 1o)) ∖ ({1o} × 1o)) |
10 | xp01disjl 8519 | . . . . . 6 ⊢ (({∅} × 𝐴) ∩ ({1o} × 1o)) = ∅ | |
11 | disj3 4457 | . . . . . 6 ⊢ ((({∅} × 𝐴) ∩ ({1o} × 1o)) = ∅ ↔ ({∅} × 𝐴) = (({∅} × 𝐴) ∖ ({1o} × 1o))) | |
12 | 10, 11 | mpbi 229 | . . . . 5 ⊢ ({∅} × 𝐴) = (({∅} × 𝐴) ∖ ({1o} × 1o)) |
13 | 1, 9, 12 | 3eqtr4i 2766 | . . . 4 ⊢ ((𝐴 ⊔ 1o) ∖ {⟨1o, ∅⟩}) = ({∅} × 𝐴) |
14 | reldom 8976 | . . . . . . . 8 ⊢ Rel ≼ | |
15 | 14 | brrelex2i 5739 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
16 | 1on 8505 | . . . . . . 7 ⊢ 1o ∈ On | |
17 | djudoml 10215 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 1o ∈ On) → 𝐴 ≼ (𝐴 ⊔ 1o)) | |
18 | 15, 16, 17 | sylancl 584 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 ⊔ 1o)) |
19 | domtr 9034 | . . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ⊔ 1o)) → ω ≼ (𝐴 ⊔ 1o)) | |
20 | 18, 19 | mpdan 685 | . . . . 5 ⊢ (ω ≼ 𝐴 → ω ≼ (𝐴 ⊔ 1o)) |
21 | infdifsn 9688 | . . . . 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 5188 | . . 3 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ (𝐴 ⊔ 1o)) |
24 | 23 | ensymd 9032 | . 2 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 1o) ≈ ({∅} × 𝐴)) |
25 | xpsnen2g 9096 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
26 | 6, 15, 25 | sylancr 585 | . 2 ⊢ (ω ≼ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
27 | entr 9033 | . 2 ⊢ (((𝐴 ⊔ 1o) ≈ ({∅} × 𝐴) ∧ ({∅} × 𝐴) ≈ 𝐴) → (𝐴 ⊔ 1o) ≈ 𝐴) | |
28 | 24, 26, 27 | syl2anc 582 | 1 ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 1o) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ∅c0 4326 {csn 4632 ⟨cop 4638 class class class wbr 5152 × cxp 5680 Oncon0 6374 ωcom 7876 1oc1o 8486 ≈ cen 8967 ≼ cdom 8968 ⊔ cdju 9929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7877 df-1st 7999 df-2nd 8000 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-dju 9932 |
This theorem is referenced by: pwdjuidm 10222 isfin4p1 10346 canthp1lem2 10684 |
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