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Mirrors > Home > MPE Home > Th. List > unsnen | Structured version Visualization version GIF version |
Description: Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.) |
Ref | Expression |
---|---|
unsnen.1 | ⊢ 𝐴 ∈ V |
unsnen.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
unsnen | ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsn 4720 | . . 3 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
2 | cardon 9987 | . . . . . 6 ⊢ (card‘𝐴) ∈ On | |
3 | 2 | onordi 6487 | . . . . 5 ⊢ Ord (card‘𝐴) |
4 | orddisj 6414 | . . . . 5 ⊢ (Ord (card‘𝐴) → ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅ |
6 | unsnen.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
7 | 6 | cardid 10590 | . . . . . 6 ⊢ (card‘𝐴) ≈ 𝐴 |
8 | 7 | ensymi 9035 | . . . . 5 ⊢ 𝐴 ≈ (card‘𝐴) |
9 | unsnen.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
10 | fvex 6914 | . . . . . 6 ⊢ (card‘𝐴) ∈ V | |
11 | en2sn 9077 | . . . . . 6 ⊢ ((𝐵 ∈ V ∧ (card‘𝐴) ∈ V) → {𝐵} ≈ {(card‘𝐴)}) | |
12 | 9, 10, 11 | mp2an 690 | . . . . 5 ⊢ {𝐵} ≈ {(card‘𝐴)} |
13 | unen 9084 | . . . . 5 ⊢ (((𝐴 ≈ (card‘𝐴) ∧ {𝐵} ≈ {(card‘𝐴)}) ∧ ((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅)) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) | |
14 | 8, 12, 13 | mpanl12 700 | . . . 4 ⊢ (((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
15 | 5, 14 | mpan2 689 | . . 3 ⊢ ((𝐴 ∩ {𝐵}) = ∅ → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
16 | 1, 15 | sylbir 234 | . 2 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
17 | df-suc 6382 | . 2 ⊢ suc (card‘𝐴) = ((card‘𝐴) ∪ {(card‘𝐴)}) | |
18 | 16, 17 | breqtrrdi 5195 | 1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∪ cun 3945 ∩ cin 3946 ∅c0 4325 {csn 4633 class class class wbr 5153 Ord word 6375 suc csuc 6378 ‘cfv 6554 ≈ cen 8971 cardccrd 9978 |
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 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-ac2 10506 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 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-pred 6312 df-ord 6379 df-on 6380 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-er 8734 df-en 8975 df-card 9982 df-ac 10159 |
This theorem is referenced by: (None) |
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