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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 3603, prprc1 3601, and prprc2 3602. (Contributed by Jim Kingdon, 16-Sep-2018.) |
Ref | Expression |
---|---|
prexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq2 3571 | . . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵}) | |
2 | 1 | eleq1d 2186 | . . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V)) |
3 | zfpair2 4102 | . . . . 5 ⊢ {𝑥, 𝑦} ∈ V | |
4 | 2, 3 | vtoclg 2720 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → {𝑥, 𝐵} ∈ V) |
5 | preq1 3570 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵}) | |
6 | 5 | eleq1d 2186 | . . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V)) |
7 | 4, 6 | syl5ib 153 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑊 → {𝐴, 𝐵} ∈ V)) |
8 | 7 | vtocleg 2731 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → {𝐴, 𝐵} ∈ V)) |
9 | 8 | imp 123 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 Vcvv 2660 {cpr 3498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 |
This theorem is referenced by: prelpwi 4106 opexg 4120 opi2 4125 opth 4129 opeqsn 4144 opeqpr 4145 uniop 4147 unex 4332 tpexg 4335 op1stb 4369 op1stbg 4370 onun2 4376 opthreg 4441 relop 4659 acexmidlemv 5740 pr2ne 7016 exmidonfinlem 7017 exmidaclem 7032 sup3exmid 8683 xrex 9607 2strbasg 11987 2stropg 11988 isomninnlem 13152 trilpolemlt1 13161 |
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