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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pren2d | Structured version Visualization version GIF version |
Description: A pair of two distinct sets is equinumerous to ordinal two. (Contributed by RP, 21-Oct-2023.) |
Ref | Expression |
---|---|
pren2d.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
pren2d.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
pren2d.aneb | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
pren2d | ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pren2d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | 1 | elexd 3501 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | pren2d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 3 | elexd 3501 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | pren2d.aneb | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
6 | pren2 43542 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵)) | |
7 | 2, 4, 5, 6 | syl3anbrc 1342 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 {cpr 4632 class class class wbr 5147 2oc2o 8498 ≈ cen 8980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-1o 8504 df-2o 8505 df-en 8984 |
This theorem is referenced by: (None) |
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