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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 4627 | . . 3 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
2 | cardon 9560 | . . . . . 6 ⊢ (card‘𝐴) ∈ On | |
3 | 2 | onordi 6318 | . . . . 5 ⊢ Ord (card‘𝐴) |
4 | orddisj 6251 | . . . . 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 10161 | . . . . . 6 ⊢ (card‘𝐴) ≈ 𝐴 |
8 | 7 | ensymi 8678 | . . . . 5 ⊢ 𝐴 ≈ (card‘𝐴) |
9 | unsnen.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
10 | fvex 6730 | . . . . . 6 ⊢ (card‘𝐴) ∈ V | |
11 | en2sn 8718 | . . . . . 6 ⊢ ((𝐵 ∈ V ∧ (card‘𝐴) ∈ V) → {𝐵} ≈ {(card‘𝐴)}) | |
12 | 9, 10, 11 | mp2an 692 | . . . . 5 ⊢ {𝐵} ≈ {(card‘𝐴)} |
13 | unen 8723 | . . . . 5 ⊢ (((𝐴 ≈ (card‘𝐴) ∧ {𝐵} ≈ {(card‘𝐴)}) ∧ ((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅)) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) | |
14 | 8, 12, 13 | mpanl12 702 | . . . 4 ⊢ (((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
15 | 5, 14 | mpan2 691 | . . 3 ⊢ ((𝐴 ∩ {𝐵}) = ∅ → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
16 | 1, 15 | sylbir 238 | . 2 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
17 | df-suc 6219 | . 2 ⊢ suc (card‘𝐴) = ((card‘𝐴) ∪ {(card‘𝐴)}) | |
18 | 16, 17 | breqtrrdi 5095 | 1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∪ cun 3864 ∩ cin 3865 ∅c0 4237 {csn 4541 class class class wbr 5053 Ord word 6212 suc csuc 6215 ‘cfv 6380 ≈ cen 8623 cardccrd 9551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-ac2 10077 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-wrecs 8047 df-recs 8108 df-er 8391 df-en 8627 df-card 9555 df-ac 9730 |
This theorem is referenced by: (None) |
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