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Mirrors > Home > ILE Home > Th. List > prexg | GIF version |
Description: The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3702, prprc1 3700, and prprc2 3701. (Contributed by Jim Kingdon, 16-Sep-2018.) |
Ref | Expression |
---|---|
prexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq2 3670 | . . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵}) | |
2 | 1 | eleq1d 2246 | . . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V)) |
3 | zfpair2 4210 | . . . . 5 ⊢ {𝑥, 𝑦} ∈ V | |
4 | 2, 3 | vtoclg 2797 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → {𝑥, 𝐵} ∈ V) |
5 | preq1 3669 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵}) | |
6 | 5 | eleq1d 2246 | . . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V)) |
7 | 4, 6 | imbitrid 154 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑊 → {𝐴, 𝐵} ∈ V)) |
8 | 7 | vtocleg 2808 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → {𝐴, 𝐵} ∈ V)) |
9 | 8 | imp 124 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 {cpr 3593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 |
This theorem is referenced by: prelpwi 4214 opexg 4228 opi2 4233 opth 4237 opeqsn 4252 opeqpr 4253 uniop 4255 unex 4441 tpexg 4444 op1stb 4478 op1stbg 4479 onun2 4489 opthreg 4555 relop 4777 acexmidlemv 5872 pr2ne 7190 exmidonfinlem 7191 exmidaclem 7206 sup3exmid 8913 xrex 9855 2strbasg 12577 2stropg 12578 prdsex 12717 xpsfval 12766 xpsval 12770 isomninnlem 14748 trilpolemlt1 14759 iswomninnlem 14767 iswomni0 14769 ismkvnnlem 14770 |
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