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| Mirrors > Home > MPE Home > Th. List > enpr2d | Structured version Visualization version GIF version | ||
| Description: A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-un 7755. (Revised by BTernaryTau, 23-Dec-2024.) |
| Ref | Expression |
|---|---|
| enpr2d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
| enpr2d.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
| enpr2d.3 | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| enpr2d | ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | enpr2d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
| 2 | enpr2d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
| 3 | 0ex 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ V) |
| 5 | 1oex 8516 | . . . 4 ⊢ 1o ∈ V | |
| 6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ V) |
| 7 | enpr2d.3 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
| 8 | 7 | neqned 2947 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| 9 | 1n0 8526 | . . . . 5 ⊢ 1o ≠ ∅ | |
| 10 | 9 | necomi 2995 | . . . 4 ⊢ ∅ ≠ 1o |
| 11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ≠ 1o) |
| 12 | 1, 2, 4, 6, 8, 11 | en2prd 9088 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ {∅, 1o}) |
| 13 | df2o3 8514 | . 2 ⊢ 2o = {∅, 1o} | |
| 14 | 12, 13 | breqtrrdi 5185 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 {cpr 4628 class class class wbr 5143 1oc1o 8499 2oc2o 8500 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-1o 8506 df-2o 8507 df-en 8986 |
| This theorem is referenced by: 1sdom2dom 9283 prfi 9363 enpr2 10042 simpgnsgd 20120 2nsgsimpgd 20122 |
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